II ELEMENTS INTEGRAL CALCULUS; KEY TO THE SOLUTION OF DIFFERENTIAL EQUATIONS, AND A SHORT TABLE OF INTEGRALS. WILLIAM ELWOOD BYERLV, Ph.D., PROFESSOR OF MATHEMATICS IN HARVARD UNIVERSITY. SEdcrXD EDITION, RKVI^EV AXD EX LA IKIED. GTNN & COMPANY BOSTON • NEW YORK • CHICAGO • LONDON //., y ■ :^ . -^ . ... , ^ I -t -^i^^^-exc^^^ Entered according to Act of Congress, in the year 1888, by WILLIAM KLWOOn B-i-EKLY, in the OUlcc of the Librarian of Congress at Washington. '0 //l-f-t^ All RiaiiTS Reservkd. / / r,lNN .V COMI'ANI PklHTUKS • BOSTON • U.S.A. y^ ^^oQfe. /y/^//A^>7^/3yr^^^^^'^^i/ PREFACE, The following volume is a sequel to m)- treatise on the Differential Calculus, aud, like that, is written as a text-hook. The last chapter, however, a Key to the Solution of Ditlercntial Equations, ma^' prove of service to working mathematicians. I have used freeh' the worlcs of Bertrand, Beujamin Peirce, Todhunter, and Boole ; aud 1 am much indebted to Professor J. M. Peirce for criticisms and suggestions. I refer constantly to ni}- work on the Differential Calculus as Volume I. ; and for the sake of convenience I have added Chapter V. of that l)ook, which treats of Integration, as an appendix to the present volume. W. E. BYEliLY. C'AMUiaUCJE, 1881. 54243S PREFACE TO SECOND EDITION. In enlarging my Integral Calculus I have used freely Scljlomilch's '• Compendium der IlOliereu Analysis," Cayley's " Elliptic Functions," Meyer's " Hestininite Integrale," For- syth's " Ditferential Equations," and Williamson's "Integral Calculus." The chapter on Theory of Functions was sketched out and in part written by Professor B. O. Peirce, to whom I am greatly indebted for numerous valuable suggestions touching other portions of the book, and who has kindly allowed me to have his Short Table of Integrals bound in with this volume. \V. K. BYEULY Cambkiuue, 1888. ANALYTICAL TABLE OF COiXTENTS. CHAPTER SYMBOLS OF OPEIJATION. Article. Page 1. Functional symbols rci;;irck'(! ;is syjH/jo/.'? o/op^m /X + HI / 71. .Application of the liidxution Formulas of Chapter V. to irra- tional forms. Examples CI 72. A function rendered irrational tliroufili the presence under the ratlical si;:n of a polynomial of higher decree than the second cauuut ordinarily he integrated. Elliptic Integrals . Ii2 CHAPTER VII. TRANSCKXliKXTAL Kl'.NCTIOXS. 73. Use of the method of InUgmtion by Parts. Examples . . . . G3 74. Hediiction F(»rmulas for .siu''a: and cos-'j-. Examples (54 75. Integration of (sin-' J-)" (?r. Examples (15 76. r.se of the methoil of Integration by Substitution CO 77. Substitution of r=tan^ In Integrating trigonometric forms . 67 78. Integrathm of sin"/cos''/-.(?j-. E.vamples C8 79. Kcduction formulas for ftan-x.dj; and i-—^- Examples. . 69 J Jtan'-z CHATTER VIII. r»KKINITK INTK.(iKAI.S. 80. Dednltlon of a (hjlniu- intnjral as the Hndt of a sum of Inllnlteslnmls 7] M. Conij)ntallon of a tieflnlte integral as the limit (.f a sum. Illus- trative exiin>|)les. Examples 72 82. Usual nieth.xl of obtaining the value of a dellnite integnil. Cautlou couceruiiig multiple-valued luucUous. Examples . 76 TABLE OF CONTENTS. IX Article ,^ Tuk-f. 83. Consideration of tlio nature of the value of | fx . dx wlien fx becomes infinite for a; = a, or x = 6, or fur .some value of x between a and 6. Illustrative examples 80 84. Maximum-Minimum Theorem. Test that must be satisfied in order that \ fx .dx may be finite and determinate if fx is infinite for some value of x between a and b. Illustrative examples. Examples 82 fx . dx shall be finite 85. Meaning of | fx . dx. Condition that | and determinate 88 86. Proof that certain important definite integrals of the form fx . dx are finite and determinate. Examples .... 89 I obtained 87. Application of reduction formulas to definite integrals. Examples 01 88. Application of the method of integration by sribstitution to definite integrals. Illustrative examples. Example . . . O'j 89. Differentiation of a definite integral. Examples 9G 90. Many ingenious methods of finding the values of definite inte- grals are valid only in case the integral is finite and de- terminate 100 91. Integration by development in series. Examples 100 92. Values of | log sina;.(?x, | e *V?x, I " '-dx c/o • «/o »/o ^ by ingenious devices. Examples 102 93. Differentiation or integration with respect to a quantity wliith is independent of x. Examples 10.') 94. Additional illustrative examples. Examples 10(1 95. Introduction of imaginary constants HW 96 The Gamma Function 109 97. Table giving logr(H) from n= 1 to n = 2. Definite integrals expressed as Gamma Functions Ill 98. The Beta Function. Formula connecting the Beta Function with the Ganinia Function. Value of r(.]) 113 99. More definite integrals expressed as Gamma Functions. Examples Hi INTEGKAL CALCULUS. rHAl'TF.U IX. i.k.V(;tiis or iikves. Article. P«ge- KM). Formulas for slnr ami cos r In terin.s of the lenjrth of tin arc 117 101. The e(|uation of the Crt/<'Hrtry obtaiiu'd. E.vaniple . . . .117 102. The cijnntion of the 7'r7ry/(/. E.xample 122 105. Another method of rectifying the Cycloid 123 IOC. Ucctillcation of the Epiryrhiid. Examples 12"5 107. Arc of the A7/i;w. Auxiliary an^le. Example 124 108. Lenfrth of an arc. I'olar coordinates 125 109. Equation of the Logarithmic. Spiral 125 110. Kectillcation of the Lofrarithmic Spiral. Examples . . . . 120 111. Rectification of the tVin/joiWe 12() 112. Itirolutes. Illustrative example. Example 127 ll.'l. The i/irfi/wr^ of the Cycloid. Example 129 114. Fiitrinsic equation of a curve. Example 130 115. Intrinsic equation of the Epicycloid. Example i;U Ufi. Intrinsic equation of tlie Lofraritlnnic Spiral 182 117. Method of obtaininj? tlie intrinsic equation from tlie equation in rectanifular coordinates. Examples 1.32 118. Intrinsic e<|uation of aiwi«/Mi-t\veen tlic tructrix and the axis. Example .... 143 TABLE OK CONTKXTS. Xl Article. Page. 12',). Area between a curve and its asymptote. Examples . . .144 130. Area of circle obtained by the aid of an au.xiliary anjjle. Ex- amples 145 131. Area between two curves (rect. coor.). Examples .... 146 132. Areas in Polar Coordinates. Examples 147 133. Problems in areas can often be simplified by transformation of coordinates. Examples \',0 134. Area between a curve and its evolute. Examples loO 135. Holditch's Theorem. Examples 151 ISG. Areas by a double intrr/ratiou (rect. coor.) 153 137. Illustrative examples. Examples. l.')4 138. Areaahy a double intei/ration (polar coor.). Example . . .155 CHAPTER XI. ARE.VS OK SUHFACES. 139. Area o{ a surface of 1-evoIution (rect. coor.). Example . . . 157 140. Illustrative examples. Examples 158 141. Area of a surface of revolution by transformation of coordi- nates. Example 159 142. Area of a s?(r/«ce o/rewoZ?/??. (polar coor.). Examples . . 161 143. Area of a cylindncal siirfacp. Examples 161 144. Area of any surface by a double integration 104 14."). Illustrative example. Examples 107 \\C^. Illustrative example requiriug transformation to i)olar coordi- nates. Examples 109 CHAPTER XII. VOI.r.MKS. 147. Volume by a single integration. Example 172 US. Volume of a co«oiV/. Examples 173 14'J. Volume of an ellipsoid. Examples 174 150. \o\\xmeoi a solid of revolution. Single integration. Exam- ples 175 151. \o\\\mc oi a solid of revohition. Double integration. Exam- ples 176 152. \o\\\me of a solid of revolution. Polar formula. Example . 178 153. Volume of any solid. Triple integration. Rectangular coor- dinates. Examples 179 154. Volume of any solid. Triple integration. Polar ccxirdinales. Examples 183 INTEGIiAL CALCULUS. CHAriKK XIII. CKNTKKS UK tiUAVITY. Article. Page. 155. Centre of Gravity (U'llned .184 150. General lonmiliL"^ for the eoorilinates of the Centre of Gravity of any mass. Example 184 157. Centre of Gravity of a homogeneous body 186 158. Centre of Gravity of a ;)/rtHe arprt. Example.s 186 159. Centre of Gravity of a /jomf)(;rc>ietiHs »o/i(i of revolution. Ex- amples 189 160. Centre of Gravity of an arc ; of a surface of revolution. Ex- amples . 191 161. Properties of Guldin. Examples 192 ClIAPTEU XIV. LINE, Sl-UKACK, AND Sl'ACK INTEORALS. IC,2. Point function. CV>h. Value of a line, surface, or space integral independent of the manner in which the line, surface, or space is l)rok('n up Into inllnitesimal elements 195 166. Geometrical representation of a line-integral along a plane curve, and a surface-integral over a i)laue surface . . .197 367. Moments of inertia. Examples ' 197 168. Uelatlon between a surface-integral over a plane surface, and a line-integral along the curve bounding the surface. Ex- ample; 'jcO 169. Illustrative example. Examples 202 170. Another form of the relation established in .\rt. 168. Example 203 171. Relation between a si)ace-lntegral taken throughout a given space and a surface-Integral over the surface bounding the spa<-e. Example 203 172. Illustrative example. Example . . 205 T^UiLl-: OF CONTENTS. XXii CHAPTER XV. MEAN VALUE AND I'KOBABILITY. Article. Page. 173. References 206 174. Mean value of a continnonsly varying quantity. The mean dis- tance of all the points of the circumferences of a circle from a fixed point on the circumference. The mean di.-) 222 182. Actual computation of f(—, -\ and f(—. -").... 223 183. Landen's Transformation. Reduction formula by which we can increase the modulus and diminish the amplitude of an Elliptic Integral of the second class. A method of com- puting E{k,p) 220 184. A reduction formula for diminishing the modulus and in- creasing tlie amplitude of an Elliptic Integral of the second class. A secoud method of computing E {k, . An Elliptic Integral of the llrst or second class, whose anipli- tiule is greater than -, can be made to depend upon one whose amplitude is less than ^, and upon tliu correspond- ing complete Elliptic Integral 235 187. Three-place table of Elliptic Integrals of the llrst class and the secoiul class 237 18S. Addition Formulas. Functions defined by the aid of definite integrals, logs-, sin 'x, tan'' r, F(k,x), E (k,:r). Addition formula for log x 239 l.s'.i. Addition formulas for sin'. r and tiin' .r 241 i;iU. Addition formula for J^(t,a;) . . . 243 I'Jl. .Vnalogy between log^M, sin?/, tan 7(. and /^'^ (i*, ?<) . . . .245 r.»2. The Kllijidr Fttnrtions, sum, cn?(, and dn ?/. Their analogy with Trigonometric Functions. Formulas connecting the Elliptic Functions of a single (juantity 246 Ii».'5. Formulas for Elliptic Functions of (u-j-f) and (u — v) . . . 248 WW. Formulas for Elliptic Functions of 2 w 250 195. Formulas for Elliptic Functions of " 250 19G. /V')-i'o(/iV{7»/ of the Elliptic Functions. Real period, 4 A'. . ,251 197. Elliptic Functioub of a pure imaginary. Jacobi's Transforma- tion. Elliptic Functions have an imaginary period, 4 K'V—l. Table of values of Elliptic Functions having the modulus V2 253 2 198. The Elliptic Integral of the second class expressed in terms of Elliptic Functions. Addition formula for Elliptic Inte- grals of the second class 250 199. Application to the rectification of "the f.fmniscatc Examples. Bisection of the arc t)f a t|ua(lrant of the Lemniscate . . . 258 200. Kectillcation f)f the Ellipse. Examples 261 201. Use of the Addition Formula in dealing with Elliptic arcs. FngnanVa Point. Examples 262 202. Rectification of the Jli/prrhula. Examples 264 203. The simple pendulum. Examples 267 204. Direct integration of irrational Algebraic functioii.s. Examples 275 205. Transformation of Elliptic Integrals. Examples 281 206. Integration of Elliptic Functions. Examples 282 TABLE OF CONTENTS. XV CHAPTER XVII. INTRODUCTIOX TO THE TIIP:OKY OK KIXCTIOXS. Article. Page. 207. Single-vahied functions. Multiple-valued functions .... 283 208. Importance of the graphical representation of iniagliiaries. Complex quantity 283 209. When a complex variable is said to varj- continuouslj'^ . . . 284 210. A continuuHs function of a complex variable. Critical values 284 211. Criterion that a function shall have a determinate derivative. Monoyenic functions 285 212. Any function involving z as a whole is a monogenic function 288 213. Conjugate functions. Their use as solutions of Laplace's Equation. Example 288 214. Preset-cation of angles 290 215. If two paths traced by the point representing the variable have a common beginning and a common end, and do not enclose a critical point, tlie corresponding paths traced by the point representing the function and having a common beginning will have a common end 292 216. Examples where the. paths traced by the point representing the variable enclose a critical point 294 217. Critical points at which the derivative of the function is zero or infinite are to be avoided. Branchpoints. Uolomorphic functions 295 218. Definite integral of a function of a complex variable defined. Such an integral is generally indeterminate, and depends upon the patli by which the point representing the variable passes from the lower limit to the upper limit of the integral 297 219. If the function is holomorphic, tlie definite integral is in gen- eral determinate 299 220. Tile integral around a closed contour, embracing a point at Avhich the function is infinite 300 221. Illustrative examples 301 222. Convcrgency of the series o])tained by integrating tlie terms of a convergent series where the separate terms are holo- morphic functions 303 22.3. Proof of Taylor's and Maclaurin's Theorems for functions of complex variables. Circle of convergence 304 224. Investigation of the convergency of varions series which arc obtained by Taylor's and Maclaurin's Tiieorems. Examples 307 inte(;i:al calculus. CHArTEU XVIII. KEY TO THE SOLL'TIOX OK UIFl-KKKNTIAL EQUATIONS. Article. Page. 226. Description of Key 312 226. Dodnition of the terms differential equation, order, degree, linear, general solution or complete primitive, singular solu- tion, exact differential equation 012 227. Examples lllnstratingr the use of the Key 314 228. Sinipliflcatlfin of ditlerentlal c(niatioiis by chan24 Ki-.Y 820 ExAMi'LEjj T-xni-.K Kr.Y 349 intectEal calculus. CHAPTER I. SYMBOLS OF OPERATION. 1. It is often convenieut to regard a runctioiial symliol as indicating an operation to be 2^(^rforme(l upon the expression u-hich is ivritten after the symbol. From this point of view the symbol is called a symbol of operation, and the expression writ- ten after the symbol is called the subject of the operation. Thus the symbol D^ in D^{x-y) indicates that the operation of dillerentiating with respect to x is to be performed upon tlie subject (x^y). 2. If the result of one operation is taken as the subject of a second, there is formed what is called a compound fumiiou. Thus logsinx is a comjMund function, and we may s|)eak of the taking of the logsin as a compound operation. 3. When two operations are so related tliat the compound operation, in which the result of performing the first on any subject is taken as the subject of the second, U-ads to the same result as the compound operation, in whidi tlie result of per- forming the second on the same subject is taken as the subject of the first, the two operations are commutative or nlali eel >j free. Or to formulate ; if fFH=Ffu. tlie operations indiratcd l»y/ and /-'are <, are distriljutive, for we know that d{u± v)=dH±dv, and that A(» ± v) = D,n ± D^v. The operation sin is not distributive, for sin(?< + v) is not equal to sin?t + sin v. 5. The comjyoxindfi of distributive operations are distribittive. Let / and F indicate distributive operations, then fF will be distributive ; for F{u ± v) = Fu ±Fv, tliercfore fF{ u ± r)=f{ Fu ± Fv) = fFu ± fFv. <). The repetition of any operation is indicated by writing an exponent, equal to the number of times the operation is per- formed, after the symbol of the operation. Thus log'''jr means log log log .r ; (J?u means dddu. In the single case of the trigonometric functions a dilferent use of the exponent is sanctioned by custom, and sin-'w means (sinu)* and not sin sin?<. 7. if in and n are irhole numbers it is easily proved that Chap. I.] SYMBOLS OF OlMCUATION. 3 Thin formula is assumed for all values of m and n, and ueija- tive and fractional exponents are intei-preted by its aid. It xa called the law of indices. 8. To find what interpretation must be given to a zero ex- ponent, let . . , . . )n = HI the forniiila of Art. 7. or, den'^ting/"»< by v\ f^v = v. That is ; a symbol of operation icith the exponent zero has no effect on the subject, and ma}' be regarded as multiplying it by unit}'. 9. To interpret a negative exponent, let m= — n in the formula of Art. 7. /-"/"«=/-« + »»=/'» = «. If we call 'fu = V, then /-" y = u. If v = l we get f~^f>t = "i and the exponent —1 indicates what we have calh-d the anti- function of /(<. (I. Art. 72.) The exponent —1 is used in this sense even with trigonometric functions. 10. When two operations are commutative and disfributii'e, the symbols which represent them may be combined precisely as if they were algebraic quantities. For they obey the laws, a(m + n) = am + an, am = ma, on which all the operations of arilhm.tic and algebra are fotmd«'d. 4 INTKCKAh CAIA'ULUS. [Aur. 10. For example; if the opfnitioii {D,+ D^) is to be performed n times in siieeession on a snlyeet », we ean expand (/>, + Z)^,)" precisely as if it were a l)in()minal, and then perform on u the operations indicated by the expanded expression. (7^ + I),y H = ( /V -f- .'3 1)/I>^ -f- 3 D, D; + Z>/)m i w. II.] liMAtilNAlilES. CHAPTER II. IMAGIXARIES. 11. An imaginary is usuall}' defined in algebra as tJip /»(7/- cated even root of a negative quantity^ and although it is clear that there can be no quantity that raised to an even power will be negative, the assumption is made that an iniauinary can be treated like an}' algebraic quantity. Imaginaries are first forced upon our notice in connection with the subject of quadratic equations. Considering the typical quadratic , , , 7 a we find that it has two roots, and that these roots possess eer- taiu important properties. For example ; their sum is —a and their i^roduct is h. We are led to the conclusion that every quadratic has two roots whose sum and whose product are simply related to the coefficients of the equation. On trial, however, we fiud that there are quadratics having but one root, and quadratics having no root. P'or example ; if we solve the ccjuation ar-2r+ 1 =0, we find that the onh" value of .»• which will satisfv it is unity ; and if we attempt to solve ar-2.f4-2 = 0, we find that there is no value of x which will satisfy the equation. As these results are apparently inconsistent with tlu- conclu- sion to which we were led on solving the general equation, we naturally endeavor to reconcile them with it. The difficulty in the case of the cipiation wliidi has hut one 6 intk(;hal rAiXTLUS. [Akt. 12. root is easily ovcroomo by rc-xanling it as liaviiiir two e(iual roots, riiiis wt- can say that oacli of the two roots of the equation is equal to 1 ; and there is a decided advantairc in looking at the f chaufjing the sign of the subject of the op)eration. Thus in a\l — \ the symbol V — 1 incUcates that an operation is perfornuMl upon a which, if rci)eated. will change the sign of a. That is, a(V-l)'-= -a. Chap. II.] IMAGINARIES. 7 From this point of view it would be more natural to write the symbol before instead of after the subject on which it oi)i'rates, (V — 1)« instead of aV — 1, and this is sometimes done; but as the usage of mathematicians is overwhelmingly in favor of the second form, we shall employ it, merely as a matter of con- venience, and remembering that a is the subject and the V — 1 the symbol of operation. 14. The rules in accordance with which we shall use our new sjTiibol are, first, a V^ + /j V^l = (a + h) V^l . [1] In other words, the operation indicated by V — 1 is to be dis- tributive (Art. 4) ; and second, aV^l=(V^l)a, [2] or our s3Tnbol is to be commutative with the symbols of quantity (Art. 3). These two conventions will enable us to use our symbol in algebraic operations precisely as if it were a quantity (Art. 10). When no coeflicient is written before V — 1 the coellicieut 1 will be understood, or unity will be regarded as the subject of the operation. 15. Let us see what interpretation we can get for powers of V — 1 ; that is, for repetitions of the operation indicated by the svmbol. (^rr-i)o=i (Art. 8), (V^"l)-= -1, by definition (Art. 13), {yf^^iy = ( V"^ )- V^n = - V^l , by definition, (V^l)^=iV^n = v^^, (^/^i)«=(V'^)'^ =-1, and so on, the values V— 1, —1, — V— 1, 1, occurring in cycles of four. 8 INTEGRAL CALCULUS. [AuT. 16. 10. The (Icliiiition wo liavi- '/iwn for the square root of a ney the usual method, we get x=l ± V^=a; V^4 = 2V^1, by Art. 13 [1] ; hence .r = 1 + "i V — 1 or 1 — 2 V — 1 . On substituting these results in turn in the equation (1), per- forming the o[)erations l)y the aid of our conventions (Art. 14 [1] and [2]), and interpreting (V — 1)" by Art. 15, we find that they botli satisfy the ecpiation, and that they can therefore be regarded as entirely analogous to real roots. We find, too, that their sum is 2 and that their product is 5, and consequently that they bear the same relations to the eoeflicieiits of the eipiation as real roots. 17. An imaginary root of a (juadratic can always l)e reduced to the form a + b V — 1 where a and h are real, and this is taken as the general type of an imaginary ; and i)art of our work will l>e to show that when we sulyect imaginaries to the ordinary functional operations, all our results are reducible to this typical form. If two imaginaries a-+-iV— 1 and r-\-(l\/—{ are equal, a nuist be ecpud to r,.and h nuist be e(pial to tl. Kor we have a -f ^V— 1 = f -|- rfV— 1. Tlierefore o — c =(f/ — 6)\/— 1, or a real is ecjual to an imaginary, unless ii — ,- = = rl — h. Sinc-e obviously a real and au imaginary cannot be ecpial, it follows that a = c and b = d. ClIAP. II.] IMAGINAltlES. 9 18. We have defined V — 1 as the symbol of an operation whose repetition changes the sign of the sulyect. Several different interpretations of this operation have been suggested, and the following one, in which every imaginary is graphically represented by the position of a point in a plane, is commonly adopted, and is found exceedingly useful in suggest- ing and interpreting relations between ditl'erent iuiaginaries and between imaginaries and reals. In the Calculus of Imaginaries, a-\-b V— 1 is taken as the general symbol of quantit}-. If b is equal to zero, a + b V — 1 reduces to a, and is real; if a is equal to zero, o + 6 V — I re- duces to 6 V — 1, and is called a pure imaginary. a + b-\l — \ is represented by the position of a point referred to a pair of rectangular axes, as in analytic geometry, a being taken as the abscissa of the point and b as its ordinate. Thus in the figure the position of the point P represents the imaginary a + b V — 1 . If 6 = 0, and our quantity is real, P will lie on the axis of X, wliich on that account is called the axis of reals ; if a=0, and we have a jmre imaginary, P will lie on the axis of F, which is called the axis of j)ure imaginaries. It follows from Art. 17 that if two imaginaries are equal, the points representing them will coincide. Since a and aV — 1 are represented by points eciually distant from the origin, and lying on the axis of reals and the axis of pure imaginaries respectively, we may regard the operation indicated by V— 1 as causing the point representing the subject of the operation to rotate about the origin through an angle of 90°. A repetition of the operation ought to cause the point to rotate 90° further, and it does ; for and is represented by a point at the same dist.inif from tin- 10 INTEGRAL CALCFLUS. [Art. 19. nrijjin as a, and lying on tlic opposite side of the origin ; again repeat the operation, and the point lias rot-ateil '.MP further; repeat again, and the point has rotated through 3G0°. We see, then, that if the subjeet is a real or a pure imar/iium/ the effect of performing on it the operation indicated by V — 1 is to rotate it about the origin through the angle 1)0°. We shall see later that even when the subject is neither a real nor a pure imaginar}-, the etlect of op«'rating on it with V — 1 is still to produce the rotation just described. lit. The .s»m, the product, and the quotient of an}' two imagi- naries, a -|- 6 V — 1 and c + d V — 1 , are imaginaries of the typi- cal form. a + b^/^ + c + d^T^ =a + c + (/y + fOV^. [1] (a + iV^) (c + d V^ ) = (*o - bd + {be + ad)'\r^. [2] „-(-?,V3T ^ (a-|-/jV3i) (c-d^T^) ^ ac+bd+{bc-ad)\r^ c + dyl'-i (c+ fZ\Cr[ ) (c _ fZ V ^ ) <-•" + ff"' ac + bd be — ad , All these results are of the f the anfuvieut of the imaginary. CiiAP. II.] IMAGINARIES. 11 The figure enables us to establish very sunple relations between x, y, r, and <^. [•-^] a;=reos«^, | P^, y=?-sin<^; j <^=tan-^|. j 3J + .vV— 1 = ?'cos<^ + (V — l)rsin<^ = r(cos<^+V^.siu may have any one of an infinite number of values dirtering by multiples of tt. In practice we always take the positive value of ?•, and a value of which will bring the point in question into the right quadrant. In the case of any given imaginary then, r can have but one value, while may have any one of an infinite number of values differing by multiples of 27r. The modulus r is sometimes called the absolute value of tbe imaginary. / Examples. (1) Find the modulus and argument of 1 ; of V — 1 ; of — 4 ; of-2V^^; of 3 + 3 V^; of 2+4 V^; and express each of theso^ quantities in the form r(cos(cos(/)2 + V— 1 . sin! cos <^... — sin 0, sin 0o -|- V — 1 ( sin 0, cos 0. + cos , sin ., — sin , sin 0^ = cos(0, -|- <^._,) , sinc/>|Cosc/)._, -f eoS(/),sin(^^ = sin((^, -f- 0^) by Trigonometry ; hence r, (cos<^, 4-V— l.siuf/),) r, ((•os.^j+ V^. sin02) = r,r,[cos(r^, + 4>,) -f- V-i. sin(<^, + 0,)], Chap. II.] IMAGINARIES. 13 ;uk1 our result is in the t^'pical form, fir., hoing the moduhis and 01 + 4>-> the argument of the product. If each lactor has the modulus unit}', this theorem enaliles us to construct ver}' easily the product of the imaginaries ; it also enables us to show that the interpretation of tlie o[)eration V— 1, suggested in Art. IS, is perfectl}' general. Let us operate on any imaginary subject, r(cosc/)-f V— 1. sin^), with V^, that is, with 1 f cos ^ + V^ . sin - ) . The modulus r will be unchanged, the argument <^ will be in- creased by ^, and the effect will be to cause the point rei)re- senting the given imaginary to rotate about the origin through an angle of 90°. 23. Since division is the inverse of multiplication, ?-i(cos<^i -f V — 1. sin^i) -=- r.2{cos<{>2+ V— l.sin<^2) will be equal to -i [cos (<^i - <^2) + V- 1 . sin (<^, - (^,)] , '2 since if we multiply this by ?-2(cos<^2+ V— 1. sinews)' according to the method established in Art. 22, we must get ri(cos<^i + V— l.sin<^,). To divide one imaginary by another, toe have then to take the quotient obtained by dividing the modulus of the Jirst by the modulus of the second as our required modulus, and the argu- ment of the first viinus the argument of the second as our uetv argument. 24. If we are dealing with the product of n equal factors, or, in other words, if we are raising r(cos«^ -h V— 1. sin) . [1] If r is unity, we have merely to multiply the argument by ??, without changing the modulus ; so that in this case increasing the exponent l)y unity amounts to rotating the point represent- ing the imaginary through an angle equal to <^ without changing its distance from the origin. 25. Since extracting a root is the inverse of raising to a power, 7[/-(cos<^ + V^.sinc/))] = 7rfcos^-f V^.sin-j; [1] for. bv .\rt. 24, ^/cos- + V-l.sin^j = r(cos<^-|- V- i.sin^). EXAMI'LK. •^-^how that Art. 24 [1] holds even when n is negative or fractional. 26. As the modulus of every quantity^ positive, negative, real, or imaginary, is j)ositive^ it is always possible to tind the modulus of any required root ; and as this modulus must be real and positive, // can ncvcr^ in any given example, have more than one value. We know from algebra, however, that every equa- tion of the 7tth degree containing one unknown has n roots, and that consequently every number must have n nth roots. Our formula. Art. 2;') [1], appears to give us but one j*th root for any given (juantity. It must then be incomplete. We have seen (Art. 20) that while the modulus of a given imaginary has but one value, its argument is indeterminate and may have anyone of an infmite number of values which dilfer l)v multiples of 2ir. If 0u is one of these values, the full form of Chap. II.] IMAGINARIES. 16 the imaginary is not ?-(cos h -^ — ' \- o — 1 \-in — \) — , n n n n n n n n n n n 11 as arguments of our ?ith root. Of these values the first h, that is, all except the last two, coiTespond to different points, and therefore to different roots ; the next to the last gives the same point as the first, and the last the same point as the second, and it is easily seen that if we go on increasing m we shall get no new points. The same thing is true of negative values of m . Hence we see that eveinj quantity^ real or imaginary, has n distinct nth roots, all having the same modulus, but with argu- ments differing by multiples of ^^. 27. Any positive real differs from unity only in its moddhis, and any negative real differs IVom —1 only in its modulus. All the 7ith roots of any number or of its negative may bu obtaiued 16 IN rK(;i:Ai, cAi.crLUS. [Akt. 27. by inulliplyiiig the //Ih roots ol' 1 or of —1 In the iv:il positive 7o+ V— l.sino), 111 = ^i(eos^i-f- V— l.sin^i), (fee, and suppose that the series Iio+Iii+R2 + + ^^n + is convergent ; then will the series Mq-I- "i+ ^h-\- be convergent. The series JiQ-\- Iii-\- is a convergent series composed of positive terms ; if then we break up this series into parts in any way, each part will have a definite sum or will approach a defi- nite limit as the number of terms considered is increased in- definitely. The series Wq -f- », -|- "^ -f- "„ -f can l)e broken up into the two series i?ocos*o-|- 7?,cos, -I- 7^oCos*2 + + -'?,.fos„-|- (1) and ^/^{Ros\n% + RiHin^i + R.jsm2 -\- + /i'„sin*„-|- ). (2) (1) can be separated into two parts, the first made up only of positive terms, the second onl>' of negative terms, and can therefore be regarded a.s the difference between two series, each consisting of positive terms. Each term in either series will be a term of the modulus series li^ + Ri -\- R2 + multiplied by a quantity less than one, and the sum of n terms of each series will therefore approach a definite limit, as n increases indefi- nitely. The series (1), then, which is the abscissa of the point representing the given imaginary series, has a finite sum. Chap. II.] IMAGIXARIKS. 19 In the same wa}- it ma^- l>e shown tliat the cocfrieieiit of V— 1 in (2) has a finite sum, and this is the ordinate of the point representing the given series. The sum of n terms of tlie given series, then, approaches a definite Umit as ?i is increased indefi- nitely, and the series is convergent. 31. We have seen (I. Art. 133 [2]) that ^ = ^+I + ll + i7 + l7 + f] when X is real, and that this series is convergent for all values of .r. Let us define e', where z = x-j- y^f — l, by the series «'=i+Y+|i+|^+|^+ m This series is convergent, for if ^ = /-(cos <^ + V — 1 . sin <^) the series ^ 1+-+— +— +— + i2!3:4: made up of the moduli of the terms of [2] is convergent by I. Art. 133, and therefore the value we have chosen for e' is a determinate finite one. Write x-\-y\/—l for z, and we get c^+vv^^l I •^•+W^l I (■x-+W~l)- , (a-+.W~l)' I |-3-| The terms of this series can be expanded by the Binuiiiial Theorem. Consider all the resulting terms containing any given power of X, say x^ ; we have ^' n I ^'^^^ I (!'^~^y- I (W~i)'^ I , (.yV^n" , y pr ^ 1 ^ 2! "^ 3! ^ ^ u\ ^ " or, separating the real terms and the imaginary terms, j>r 2! 4! g: ' p\ 3 ! o ! / ! 20 INTi:(iKAL CALCULUS. [AUT. 32. or — (cos v+ V^. sill'/). hv I. Art. 134. P- Giving ]) all valiu-s IVoiii (» to x wc get e'^»-'^=(cosy + V-l.siu//)(l + Y + ^ + |^ + j^+ ) = c' ( cos V + V — 1 . sin V ) , [4] which. In- the way. is in one of our typical iiuas^inary Ibrnis. If..= 0. in [4], we get (-y v^-> = cos »/ + V— 1 . sin ?/, [,'»] which suggests a. new way of writing our typical iniaginar}' ; namely, r(cos(f> + V— I. sin<^) = ir'^'K .32. We have seen th.nt let us see if all iniagiiuiry powers of e obey the law of indices; that is, if the equation e"e''=e'' + ' [1] is universally true. Let n = .<-, + 7, V— 1 and r = .7V 4- 1/.^ V — 1 , then <»"= e'l + Wi ^-' = f^i (cos?/, -|- V— 1 . sin Vi) , g. ^ px, + y, v/TT _ ,,xj(cos ?/2 + V— 1 . sin//.,) , e"e'' = e^ic^j [cos (,v, + //,,) + V— 1. sin (?/, + ?/,,)] = e*. + ^-^ [cos (//, -f- ?/,) + V^ . sin {>/, + //,) ] an = taii"'?-^ ; X whence logz = 4^ log (.r + f) + V — 1 . tan"' ^. [2] Each of the expressions for log z is indeterminate, and repre- sents an intinite number of values, dilfering l)y multiples of 27rV^. This indeterminateness in the logarithm might have l)een ex- pected a jyriori, for = cos Z TT +V-l.sin2 7r= 1, by Art. 31 Hence, adding 2 7rV— 1 to the logarithm of any (piantity will have the effect of multiplying the quantity by 1, and therefore will not change its value. Example. Show that if an expression is imaginary, all it,s logarithms are imaginary ; if it is real and positive, one logarithm is real and the rest imaginary ; if it is real and negative, all are inuigiiuiry. ^^ INTKGKAL (ALCrLUS. [Aur. 34. Trifjonometric Fidictions. 34. n-isn-al, Z^ T" y"! I'vl. Art. l;;i. ^I 4! G .'4! G!^ M tiR' .scries of the moduli, /-■• <-' ..7 y- *•■' .•« 2 ! 4 ! ^ G ! ^ se I.e. [1] and [2] are eovergent. We .shall take them as defi- nitions of the sine and cosine of an imajrinaiy. KXAMI'I.K. ^ From th(. formulas of Art. ;51, and from Art. 34 \ show that '- 1] and [2], smz, sin^, for all values of «. 35. From the relations <^''^"' = ^'OS2 + V^. sin 2, e-'^' =eos2- V^..sin2, 2 ' [1] Sm2= ^ r-,^ Tor all values of^. - v i Chap. II.] Let cos(.r-}-?/V— 1) = IMAGINARIES. z = x + y\J — \. 23 + e- ■.^/^l + y _ (cos.r+ V— 1 . sina;)e~''+ (cos a;— V— 1 . sin a;)e* 2 by Art. 34, Ex., = cosx — ■ — V — l.smx' [3] In the same wa^' it niav be shown that . . ^1 — rN _(cos.x--f- V— 1. sina;)e »'— (cosa; — V — l.sina;)( H^x+j/ V — i;— -—iz 2V-1 e" + e- + V-1. e^ - e-*' If 2 is real in [1] and [2], we have cos X = + e- [i] ,a,. = _l ^ V-1. If z=7/ V — 1, and is a pure imaginar}', e* + e- 1 = sm?/ V^ = V:^!; [5] [6] whence we see that the cosine of a pure imaginary is real, while its sine is imaginary. By the aid of [5] and [/ V— 1) = sinx-cosy V^ -I- cosxsin// V— 1. [h] 24 LNTKiiKAL CALCl'LUS. [Akt. 36. EXAMI'LES. (;) From [1] and [2] sIk.w Uiat siir2+ cos^j; = 1. 1/(2) rrovi- tliat cos ( II -f v) = COS a t'O.s V — sin n sin /', sill {n + v) — sin? + V — 1. sin<^) + rdcf){— sin +V— 1. cos eft) = {dr + rV^.dcji) (cos + V^l . sin <^) . [2] 38. Let ns now consider the dilferentiation of 2", e', logz, sinz, and cosz. Let z= r (cos <^ + V — 1 . sin (/>) , then 2"* = r'"(cosm<^ + V^. sin?u^), by Art. 24 [1] ; dz'" = ?»)•"'"' r/;- (cos ?«(^ + V— 1. sin??i^) + mr^dcf: {— sin ?n (/> + V — l.COS/«(^), ^2"= ?«)•"'"' [cos (»i — 1) <^ + V — 1. sin (/» — !) <^] (cos(^ + V^.sinf/))f//- + ?» /•"• [cos ( ?» — 1 ) + V^ . sin (^) V^ . (7<^, dz" = ?>(?•'""' [cos (?H — 1) <^ 4- V— l.sin (m— 1) <^] {dr + rV — 1 .cZ(^) (cos+ V— 1. sin0). dz'" = mz"'-' dz, [ 1 ] 1 'V Art . 3 7 [2] , dz [■^1 and a power of :in imaginar}' varial)le has a single di-rivativc. 26 INTEGRAL CALCULUS. [Art. 39. 39. If 2=a;-|-t/V^, e' = e'(cos2/+V^..sin?/), bj- Art. 31 [4j, de' = e'c/x(cosy-f-V— 1. sin?/) + (f{— ainy + V^.cos?/)f7?/, de* = e*(cosy+'\/ — l. siny) (cte+V^. cly), de'=e'dz, [1] dz Show that [2] Example. da' = a' lo-^i 2 V^ e* ^i_^,-.^--I V-l.rfz tfe, dsinz = cosz.rfz. by Art. 35 [2], by Art. 35 [1], [1] cosz = ■ CiiAP. II.] IMAGINARIES. 27 rfC0S2 = ; V — l.rfz = — ffe, f?cos2;= — sinz.dz. [2] I I 42. AYe see, then, that we get the same formulas for the dif- ferentiation of simple functions of imaginaries as for the dif- \ ferentiation of the corresponding functions of reals. It follows i that our formulas for du-ect integration (I. Art. 74) hold when x \ is imaginar}'. HyperhoUc Functions. 43. We have (Art. 35 [.5] and [H]) j cos X V3i=£!±r and sin.rV^ = ^^^V-l, where x is real. — ^^— — is called the hyperbolic cosine of «, e^-e and is written cosh.r; and — ^ — is called the hyperbolic sine of .r, and is written sinh.r; sinhx =^-^1^-^ = — V — l.sin.x^V — 1, [1] cosh.r= — ^t— - = cos.r V— 1. [2] The hyi)erbolic tangent is defined as the ratio of sinh to cosh ; and the hyperbolic cotangent, secant, and cosecant are the re- ciprocals of the tanh, cosli, and sinh respectively. These functions, which are real when x is real, resemble in their properties the ordinary trigonometric functions. ^ 28 INTEGRAL CALCULUS. [Art. 44. 44. For example, eoslr .>• — .siiil)-.r = 1 ; [1] p^^ + 2 + e-^' for cosh- a; = — -- — , 4 and sinh^'a; = ^^t_? 4 Examples. ( (1) Prove that 1 — tauh'-.i- = sech'a;. (2) Prove that 1 — ctnh^.r = — csch^a;. i (3) Prove that sinh(x + _?/) = siiih.x* cosh?/ + eoshxsinhy. \ (4) Prove that cosh (x + ?/) = cosh x cosh // + siuh a- siuhy. 45. rtsinh.r = « = — ' ax. 2 2 rfsiuhx= coshx.dx. Examples. ( 1 ) Prove 1 d cosh x = sinli x.dx. \ dtanhx = scclvx.dx. \ dctuhx = — csch^x.c/a;. ^ 1 d sech x = — sech .r tanh x.dx. *\ dcschx = — cschxctiih x.f/x. 46. We can deal with anti-hyperbolic functions just as with anti-trijiononietric functions. To find f/sinh 'x. Let « = siuh 'x, then x = 8inhj<, dx = cosh ii.dii. IP. U.j INIAOIN ARIES. 29 cosnw cosh?f = Vl +siuh^M, by Art. 44 [IJ, cosh»= Vl +af^, 7 11 (^X ttsinh X = • ri] Vl+ar' EXAMI'I.KS. Prove the formulas 11 <^^'^' Vo;- — 1 ^ dtanh^a; = - —> I ■, 11 (Jx x^/l-x" I dcsch^a; = — x yxr + 1 47. The anti-hyperbolic funetious are easily expressed as logarithms. Let w = sinh-'.r, then 2x= e" , 2xe" = e'-"— 1, e^" — 2.te"= 1, e2"-2a-e" + .vr= 1 + ar'. e" = a- ± Vl +a^; 30 INTEGRAL CALCULUS. [Art. 48. as e" is necess.irily positive, we may reject the negative value in the second iueiul)er as impossible, and we have = a;+ Vl+ar^, n = log(a; + VT+l?), Examples. Prove the formulas » /cosh"'x=log(a;+ Vx^— 1). , .-tauh ^x = ^ log . *^ ' 1 —X 48. One of the advantages arising from the use of hyper- bolic functions is that they bring to light some curious analogies between the integrals of certain irrational functions. From I. Art. 71 we obtain the formulas for direct integration. j — '-==^i:_ = siu'^x. [1] J Vl — X- f — =:sec-'a;. p]] From Art. 4G we obtain the allied fonnulas : / — ^ — =sinh 'x = lo?(.r4- Vl-f^'). m f — ^ ^ = cosh ' X = log(.i- + Var — 1 ) . [5] CiiAP. IT] BfAGIN ARIES. 81 J 1 - ar - 1 -X [6] r dx __„..-, „_,_/! , IT -) [-] JxVl-a^ °\^ ^^' C ^^-^ c-ch-^x \0"f^ 1 u ^ -)• [8J J«Var + l "U \x- Examples. Prove the formulas ^m cinh.r = — -^"^-4- — -^ ^(2) coshx = l + |j + |j + t/(3) sin (x-\- 1/ V— 1) = sin x cosh ?/ + V— 1 cosx sinh y. i/(4) cos(x + yV— 1) = cos X cosh y — V—1 sin X sinh y. /KN 4. /I / — Tx sin 2x + V— 1 sinli 2y (5) tan(x + W-l) - cos2x + cosh2>/ ' (6) sinh (x -f- y V— 1) = sinh x cos y + V— 1 cosh x sin ?/. (7) cosh (a; + >j V— 1) = coshx cosy + V— 1 sinhx siny. /ON ^ w I /~Tx sinh 2x + V— 1 sin 2y (8) tanh (x + y V— 1) = t—^ — ; n — ^ ^ V ' ^ / cQsh 2a; + cos 2y (9) tanh ^x = ^+3 + 5 + INTEGRAL CALCULUS. [AUT. 4'J. CHAPTER TIT. QENEIIAL METHODS OF INTEGRATING. 49, We have dclinod tlio integral of an}- function of a single variable .as the function which has the given function for its derivative (I. Art. 53) ; we have defined a definite integral as the limit of the sum of a set of differentials ; and we have shown that a definite integral is the difference between two values of an ordinary integral (I. Art. 1H3). Now that we have adopted tiie ditferential notation in place of the derivative notation, it is better to regard an integral as the inverse of a differential instead of as the inverse of a derivative. Hence the integral of fr.dx will be the funct^pn whose differ- ential \s fx.d.r ; and we .shallindicatc it b}' jfx.dx. In our old nt>tati()ii we should liave indicated precisely the same function by I /> ; for if the derivative of a function is fx we know that its dilli-rential in fx.dx. 60. li fx is a continuous function of .r, fx.dx has aji integral. For if we construct the curve whose equation is y = fx, we know that the area included by the curve, the axis of X, an}- fixed ordinate, and the ordinate corresponding to the v.ariable x, has for its ditrerential ydx, or, in other words, fx.dx (I. Art. ')1). Such an area always exists, and it is a detennin.ate function of .r, except that, as the position of tiie initial ordinate is wholly arl)i- trary, the expression for the area will contain an arbitrary con- stant. Thus, if Fx is tlie area in question for some one position of the initial ordinate, we shall liave ffx.dx= Fx+C, where C is an arl)itrarv constant. H(ji -F,'F[i>] Chap. III.] GENERAL METHODS OF INTEGRATING. 33 Moreover, Fx-\-C is a complete expression for \ fx.dx ; for if two functions of .r have the same differential, the}^ have the same derivative with respect to x, and therefore they change at the same rate when x changes (I. Art. 38) ; they can ditfer, then, at any instant only by the difference between their initial values, which is some constant. Hence we see that every expression of the form fx.dx has an integral, and, except for the presence of an arbitrary constant, but one integral. 51. We have shown in T. Art. 18,3 that a definite integral is the ditference between two values of an ordinary integi'al, and therefore contains no constant. Tluis, if Fx + C is the integral of fx.dx, '^' fx.dx = Fb- Fa. £■ In the same way we shall have /z.c?2 = Fb - Fa £ and we see that a definite integral is a function of the values beticeen xchich the sum is taken and not of the variable with respect to which we integrate. Since C''fr.dx=Fa-Fb, C" fx.dx = — C fx.dx. Example. / // ^i - / <- "' fx.dx + r fx.dx = i fx.dx. , u 52. In what we have said concerning definite integrals we have tacitly assumed that the integral is a continuous function between the values l)etween which the sum in (juestion is taken. If it is not, we cannot regard the whole increment of Fx as ctiual /-^' 34 INTKGIIAL CALCULUS. [Art. 53. to the limit of the sum of the partial infinitesimal increments, and the reasoning of 1. Art. 183 ceases to be valid. Take, for example. (" ^. ^T-ll ' - -'- \lx j?=j - 1 i, by I. Art. 55 (7) and apparently J-i x~ \ xj^^i \ xj^^_i I -^ ought to be the area between the curve w = — , the But axis of X, and the ordinates corresponding to a; = 1 and x = — 1 , which evidently is not —2 ; and we 1 see that the function — is discon- or tinuous between the values x= —1 and x = l. The area in question which the definite intogi'al should represent is easily seen to be infinite, for J- 1 rr € 1, and each of these expressions increases without limit as e ap- proaches zero. 58. Since a definite integral is the difference between two values of an indefinite niiegral, what we have to find first in any problem is the indefinite integral. This may be found by in- spection if the function to be integrated comes under an}' of the forms we have already obtained by differentiation, and we are tlien said to integrate directly. Direct integration has been illus- trated, and the most important of the forms which can be in- tegrated directly have l)een given in I. Chapter V. For the sake of convenience we rewrite these forms, using the differential notation, and adding one or two new forms from our sections on hyperbolic functions. -^ Chap. III.] GENERAL METHODS OF INTEGRATING. 30 2..-/- = log- :5^Ca'dx = -^. ^ J log a ^T^\ sin.r.(?x= — cosa;. ^•H cos.r.fZa; = sinx*. '^— I tan.x-.f?.c = — log cos a;. S--j ctn a-, f/a- = log sin x. qS '^^ =sin-^r. ^^ r ^-^ =si'iA^r = iog(.v + vr+1?). d.r Vx-2-1 = cosh ^T = log (X + Va-^ — 1 ) . + .X-' . ^ r ^^•^' =taub ^T = ilog{-^^. / ^^ l-ar 1 -a; ,^ r — ^g = vers 'x. //J V2X-X2 36 INTEGRAL CALCULUS. [Art. 64. 54. "We took up in I. Chap. V. the principal devices used in preparing a function for integration when it cannot be integrated directly. The first of these methods, that of intefjration by siibstitutmi, is simplified by the use of the ditlerential notation, because the formula for change of variable (I. Art. 75 [1]), j /< = I uD^ becoming | ridx= i u—dy, reduces to an identity and is no longer needed, and all that is required is a simple substitution. (a) For example, let us find | ^Vl-f logx. Let 1 + logx = z ; then — = dz, and r^ Vl + l()g.C = Cz^dz = 2 2^ = 2 (1 _^ log.'C)^ When, as in this example, a factor of tlu quantity to be integrated is equal or proportional to the differential of some function occurring in the expression, tlie sul)stitutiou of a new variable for the function in question will generally simplify the problem. (b) Itcjuircd f— ^^. Let e' = y\ then e'dx = dy, dx _ e'dx __ dy e' + i'-' ~ e-^ +T ~ y- + l' and I —^ — = I ' •' ., = tan ' i/ = tan V. (c) Re(piiri'-"^ , r...7.. ■'•'-' , ICx.VMPLES. j'y/l-Sx-x' Vl3 (2) ( xtan'^x.dx = 1±jI tan-'x - ^x. r xdx ^ 1_+ 1 ^ ^ J {l-xf 1 -X 2(1 -x)- ( , ) (• _^^_ = - ^'27,:,— 7- + a vers-'ii\ (•'») I V2 ax — 7?. dx = '—— V">o^?^^ + — «in"' '^ ~ "' . iV iSuggestiun : Throw 2 (J.c — .Jr into tlio funn t(^ — (.r — ct)^, , ,'X' , "^ ( G ) rL±iI!!ii" ,/a; = log (X -f- sin x) . Chap. III.] GENERAL METHODS OF INTEGKATLNG. 39 (0 ; • f?.r = .rtaii-. .':>' I -t 6^^^ ^ Suggestion : Introduce '- in [)lace of a;. z = sin '.r. I) f ^^- = \ J .rClog.r)" {n- I) (log.r)"-i t) Jl2glM:£) ,?., = log., [log(loga-) - 1] 10) I " . • • = ^tan,r + logcosg. where ^J (1-.^')^ ^ ^ ° . 2% r siDa; = L^ri/// — tan '( -tan.T ah 'Ctf^ /& ^^J^67Sf(^ J 40 LNTEGKAL LALCLLUS. [AnT. oG. CHAPTER IV. RATIONAL FRACTIONS. 56. We shall now attemi)t to consider s^-stematically the methods of integrating various functions ; and to this end we shall begin with rational algebraic expressions. Any rational algebraic j)ohjnomial can be integrated immediately b}' the aid of the formula .P n + \ Take next a ration(d fraction, tliat is, a fraction wliose nu- merator and denominator are rational algebraic polynomials. A rational fraction is j^roj^er if its numerator is of lower degree than its denominator ; improper if the degree of the numerator is equal to or greater than the degree of the denominator. Sinre an improper fraction can always l)e reduced to a pol3'nomial» plus a proi)er fraction, by actually dividing the mnnerator by the denominator, wc need only consider the treatment of proi)er fractions. 57. Every projjer rational fraction can be reduced to the sum of a set of simpler fractions each of tvhich has a constant for a numerator and some poicer of a binomial for its denominator ; that is, a set of fractions any one of which is of the form — (.r — a)"* fx Let our given fniction l>c ^~. Fx Ifo, b, 0, &c.. arc the roots of tin- e(juation, Fx = 0, ( 1 ) wc have, from the Tlicorv of Equations, Fx = A {X - a) (X - b) {X - c) (2) Chap. IV.] RATIOXAL FRACTIONS. 41 The equation (1) may have some equal roots, and then some of the factors in (2) will be repeated. Suppose a occurs p times as a root of (1), b occurs q times, c occurs r times, &c., (3) then Call then and Fx = A{x- o)" (.r - by {x - c)- A{x-by{x-cy = 4>x', Fx= {x — ayx fa , (x — a)Px=0. a If we represent by f^x the quotient arising from tlie divi.sion of fx — '— d}X by X — a, we shall have 4>a fa_ fx _ (f)a fxX where — — is a proper fraction, and may be treated Fx {x — ny {x — a)''-' ^x- f\x {x — ay-'^ (fiX precisely as we have treated the original fraction. Hence ^1^^ = ^±1—. + ^%^— {x — ay-^4>x {x — ay-^ {x-ay-^x By continuing this process we shall get .fa /a f^ fp.\a fx ^ <}>a a 4>a " ^ fp-f Fx~~ {x-ay^ {x-ny-'^ {x-(iy-- .»•-" <^' 42 INTEGRAL CALCULUS. [Art. 58. f X In the same wav -^^^ can be broken up into a set of fractions *" <^x having {x — hY, (x — b)^~^, &c., for denominators, phis a frac- tion which can be broken up into fractions having (x—c)", (x — cy-^ , &c., for denominators; and we shall have, in the end, fx_ A, , A, __^ + _A_4.. ^» Fx {x-uy {x-ay-^ x-a {x-by ' (^x-by-' ' x-b where K is the quotient obtained when we divide out the last factor of the denominator, and is consequently a constant. More than this, K must be zero, for as (1) is identically true, it^nust fv be true when x= x ; but when a;= x, :i!_ becomes zero, be- Fx cause its denominator is of higher degree than its numerator, and each of the fractions in the second member also becomes zero; whence K=0. 58. Since we now know the form into which any given rational fraction can be thrown, we can determine the numerators by the aid of known properties of an identical equation. g^. J Let it be reciuircd to l)reak up ~j- — — — into .simpler fractions. ^ J \ -r j By Art. 3x-l A li , O {x-iy{x-{-\) {x-iy x-i ' x+v and we wish to determine ^1, 7?, and C. CU-aring of fractions, we have Sx-\ = A{x + \) + B{x-\)(x + ])-\-C(x-\y. (1) As this equation is identically true, the coefficients of like powers of x in the two members nuist be equal ; and we have Ii-\-C=0, A-2C=S, A-B + C=-l; CiiAP. IV.] KATIONAL FRACTIONS. 43 whence we find ^=1, B=l, C=- 1 1 A 3a!-l 1,1 1 -<» ( ^-,)H. + i) = (^i7 + .7^ - 7TT- <-' The labor of detennining the required constants can often l>e lessened b}' simple algebraic devices. For example ; since the identical equation we start with is true for all values of x, we have a right to substitute for x values that will make terms of the equation disappear. Take equa- tion [1] : 3x - 1 = ..^x- + 1) + B{x + 1) (a- -1) + C(.v -1)-. [1] Letx=l, 2 = 2 A, A=l, then 2x-2=:B (x + 1) (x - \)-\-C {x -1)- ; divide by .T - 1 , 2 = B {x -\-l) + C {x- 1). Leta;=l, 2 = 2B, B=l, then —x+l=C{x-l), C=-l. Examples. (1) Show that when we equate the coefficients of the same powers of x on the two sides of our identical e(iuation, we shall alwa^'s have equations enough to determine all our refiuircd numerators. (2) Break up 9a^ + 9a;-128 .^^^.^ shnpU.,- fractions. ix-3r{x + l) 59. The partial fractions corresponding to :uiy giMn factor of the denominator can be determineil ilirectly. 44 INTEGRAL CALCULUS. [Aur. 50. Lot US suppose that the factor in question is of tlie first degree and occurs but once ; represent it b}' x — a. ^=-^ + -f^ (I) Fx x-a^ x' ^ ^ by Art. 57, where ^« <^x = , x — a so that Fx = (x — a) (jyx. Clear (1) of fractions. fx = A^x + {x-a)f,x. (2) As (1) is an identical equation, (l') will be true for an}' value of x. Let a; = a, fa = A4>a, A = ^, (3) a result agreeing with Art. 57. Hence, to find the mmierator of the fraction corre-fpoxdivrj to a factor (x — a) of the first degree, tee have merely to strike out from the denominator of our original fraction the factor in ques- tion, and then substitute a for x in the result. If the factor of the denominator is of the ni\\ degree, there are V i)artial fractions corresponding to it. Let (x — a)" be the factor in question. fx^ A, A. I ^3 , . A fx ,,s Fx (x-a)''"^(x-a)"-'"^(.r-rt)"-2'^ "^x-ci^^x^ ' where Fx = {x—a)"x. fx Multiply (1) by (.-c — a)", and represent (x — a)"^ by *x. x =.lj + .l,(.c - a)-\-A.,{x - a)- -f- + A„{x - ay-'' x CfiaI'. IV. j RATIONAL FRACTIONS. 4j DifiVreiitiate successively both ineinlHTs of this i(U'iitit\. nn"a, 2 ! -(44 = — "'a, '3! (u-1) Although these results form a eoinplete solution of the prob- lem, and one exceedingly neat in theor}", the labor of getting the successive derivatives of *a; is so great that it is usually easier in practice to use the methods of Art. 58 when we have to deal with factors of higher degree than the first. So far as the fractions corresponding to factors of the first degree and to the highest powers of factors not of the first degree are concerned, the method of this article can be profitably combined with that of Art. 58. 60. As an example where the method of tlie last article applies well, consider S-t;-! ^A_^ B , C X (.^• — 2) (.r + 1 ) X x—'2 x- + 1 l_r '^'"^'-^ 1 =- [(.r-2)(.r+l)i = o 2' [_x{x-\-\)i-_, 6' Lx(.x--2)i.., ;3 3.r-l 1 1 . •■' 1 I 1 x{x - 2) (.c -I- 1 ) 2 a; G x - 2 3 x + 1 46 INTEGRAL CALCULUS. [Art. 6L 61. Although the theory expounded iu the preceding articles is complete and can be applied without serious diffi- culty to the case where some or all of the roots of F{x) = [Art. 57, (1)] are imaginary, there is a practical convenience in modifying the method so as to avoid the explicit intro- duction of imaginaries into the process of integrating a rational fraction. We know (Art. 28, Ex. 2) that if the denominator of our given fraction contains an imaginary factor (x—a — &V— 1)" it will also contain the conjugate of that factor, namely, (x — a + b V— 1)", and will therefore contain their product l(x — ay-\-b^Y. Moreover, since by Art. 59 the numerator of the partial fraction whose denominator is (x — a-\-b_^ — 1)^ is the same rational algebraic function oi a—b V— 1 that the numerator of the partial fraction whose denominator is (x — a — b-sJ—lY is of a + i V— 1, these two numerators must be conjugate imaginaries by Art. 28, Ex. 3. Hence, for every partial fraction of the form 1= — we shall {x — a — b'sl—l)P have a second of the form {x-a + b yj- ly Let {x-a-b V^)" = X-\- Y V^, X and Y being real functions of x ; then {x-n + b V=^)P =X-Y V=a. The sum of the two fractions A-\-B V^ A-B V^ {x — a-b^—\y {x — a+byj—iy A -{- B yj^^ A - B -sl^^ _ 2AX-\-2BY AT-f y V- 1 A'- y V- 1 [(x- ay + b^y and is a real proper fraction. Hence, Chap. IV.] RATIONAL FRACTIONS. 47 A /i« I f-pr every numerator being of lower degree than its denominator. If we take — ' ' ., , ,„-,„ and divide numerator and de- nominator by (.r — af + h"^ we shall get a fraction of the form ^- ^ ^., , ,..-, — r and R will be of the first degree and [(a: — a)- + ^^]"-i * therefore of the form L^x + -1^> and we shall have By successive repetitions of this process we can reduce to [(x-af-\-b'y'^[(x-ay-\-b'Y-'^""^ (x-af-^fj' ' Treating all the partial fractions in (1) in this way and adding the results, we shall at last reduce (1) to the form fie A,x-\-B, A^x + B^ [(x-ay + Py'4>x~ [(x-af + O'f^ [(x-ay-^b']"-' and our partial fractions are simple in form and do not involve imaginaries. The coefficients in (2) can be found by either of the proc- esses illustrated in Art. 58. 62. Let us now consider a ratlier difficult example, where it is worth while to combine all our methods. 48 INTEGRAL CALCULUS. .r^+l ,. _ 1 w r" 4- 1 \i [Art. G2. a;8-f-] =(j.4-i)(j.2-^-|-l) and x'' — x-\-l=0 has imagi- nary roots. x^ + 1 x^-\-l (x-i)(x»+iy~(x-i)(x-\-iy(x'-x-\-if - '^ I ^^ \ ^' \ <^i^ + A I ^2^-4- A ,.. ~ x-l^ {x+\r-^ .r + \^ {x-'-x+iy^ x'-j: + 1 ^ ^ ^^= U-l)(.:^'-x+l)d.= -r~*" Substitute in (1) the values just obtained, clear of fractions and reduce and we have - 9 .r« + 2 .r« — 6 X* — 8 x'^ + 8 x2 + 6 a- + 7 Divide through by x^ — 1, and we get — 9a;* + 2x''-15x2-6x-7 = ]8^A(a:^-.r + l)^+(.r + l)[C,a- + A + (C,..T + A)(^'-•i• + l)J^ Let a; = — 1, and we find Ji, = -h Substitute this value for A and reduce ; — 6 J-* — 4 a-"' — G a-^ — 12 a- — 4 = 18 (.r + 1) [ C.r 4- />; + ( C\x + A) (.r^ - .r + 1) ]. Divide by jr + 1 and expand and we get [18 C^ + 6] a"' - [18 (Ca - A) + 2] .r' + [18(r, - Ih + r,) + 8] .<• + ISCA + A) + 4 = 0. CiiA!-. IV.] RATIONAL FRACTIONS. 49 This equation must hold good whatever the value of x, whence 18 C2 +6 = 0, 18(C2-A) + 2 = 0, 18(0,- D,-\-C\) + 8=0, 18(A+A)+'i = 0, and c. = — h A = — h c\ = — h D, = Hence, x'+l 1 1 1 1 {x — \){x'-\-iy 2 a;-l 9 (.r + 1)- 6 .r + 1 _1 X 1 3 ■>■ + 2 3 " (a;^ — X + 1)^ 9 ' x"-^ — x + 1 ' (2) 63. Having shown that any rational fraction can be reduced to a sum of fractions which always come under the four forms A A Ax-\-B Ax + B (x — a)"' x — a (x — af-\-lr' [(x — «)-^ + Z--]"' it remains to show that these forms can be integrated. To find ; -' J (x — ay let z = x — a, then dz = dx, r_Adj-_ _ Cd± _ _ 1 A J {x - ay ~ " J z" ~ (n - 1) ' z'^-' {n — V) (x — ity To find C-^^, J X — a let z = x — a, then dz = dx, cn 60 INTEGRAL CALCULUS. [Akt. 63. and J^^ = Af~=A\ogz = A\og{x-a). [2] Turning back to Art. 58 (2), we find / (8x — l)dx _ r clx r dx _ C_dx__ _ _ 1 ix-lY{x^l)~J (x-iy'^J x-1 J x + l~ x — 1 -\-\os{x-l)-log{x + l)=--^ + log''~' x-1 ' °x-\-l Turning to Art. 60 (1), we have / (3x — l)dx _ rdx r dx _ r dx x(cc-2)(x + l)~ V x^Vx-2 Va: + 1 = i logx + I log(a: - 2) - I log(.r + 1). ^ , , r(Ax-{-B)dx To find ) f \2\_j2 - J (x — ay + ¥ Ax + B _ A(x — a) Aa-{- B (x - ay -\-b^~ (x- ay -\-b'^ (x- ay + b^' If we let z = (x — ay + b^, dz = 2 (x — a) dx, and /A(x — a)dx A rdz A A no i /an If we let z = x — a, dz = dx, and C( Aa-\-B)dx ... j,^ r_J^ Aa -\- B , ,z Aa -\- B , ,x — a — tan~ ^ 7 = ; tan~ b b b b Hence, r^<-^-+ fit 'J {x — aY-\-b^ A. ^. ,, . ,2n I -'^n -\r -B , .x — a _„_ = - log [(.r - ay + b'^ + ,— tan- ' — — • [3] _ . , C (Ax+B)dx To find I p7 -rj-n J [(x — ay + b^ [(•r-ay + b^y Ax-\- B _ A{x-a) Aa + 7? l{x - ay + l^'Y ~ l{x - ay + />2]» "•"[(.*•- ay + i^]" ' CiiAP. IV.] RATIONAL FRACTIONS. 61 If we let z = (x — ay + b-, ch = 2(x — a) dx, and / A(x — a) clx _ A r(f-_ A _„^, l(x-ay-{-lj^f~ 2J ^■'~ 2{ri-lf 2(?i — 1) \_{x — ay-\-b^J-^ If we let z^x — a, dz-=^ dx, and C {Aa-\-B)dx C dz I '^ can be made to depend upon j / 2_i_/2xn-i ^J *^® aid of the reduction formula [6], Art, 64, which for this special form reduces to / dz 2 (/i - 1) J? if + h'y-^ "^ 2 (n - 1)^*2 J (^2 + Z,2y.-i L J Hence f_(:l^±^H£_= A 1 /dx 1 a — g [(X - ay + ^'•^]" ~ 2 (« - 1) b' ' \_{x - ay 4- b^r-' ■ 2?l-3 r r/.r "^2(«-l)^'-J [(x-«)^ + Z''^]''-' L J /* dx A repeated use of [6] will reduce J ^ _ ,,, ^g-., to /dx _ x2 I 7,2 ' which has already been found 1. U 1 4. -1 ^~« to be 7 tan ' — ; — 62 INTEGRAL CALCULUS. [Art. 63. Turning back to Art. (52 (2), we find that r (.r-+l)^/.r ^ , rj.r_ _ ^ f dx _ f dx_ J (X - 1) (X-' + 1/ U.r-1 '''J(x + iy *Jx + l r .r d.r _ J r (3x-i- 2) dx ^ J (.r' - X + If O x'-X-\-l = ^log(.r-l)+ l--^-h^og{x + l) x — 2 ,- ^ ,2a; — 1 -ilog(.r^-x + l)-,7^V3tan-i^^ ^ ^ x' + 1 ' ^ a-3 + 1 ^ V3 Examples. r_^^3^3_^,^_ .>::_2. J x-^ — 4 X + 2 ^ J X- + 1 ,. r dx ,, (.«-!)- 1 , _,2.-r + l 4) I =X\oii^-^ tun ' ■ — ^ Jur'-l ^ ^.tr^ + x- + l V.3 V3 5) = — - tan ' - + -T-T log J iC — x* 2(r u 4«'* a—x ^ J (a^'4-l)(.r-' + x + l) - ^ .r' + l V3 V3 7) r_^^f_ = ilog^i^+^tan-'-^. '^ .1 j,4 4.-ar'_2 ^ ^a' + l 3 V2 ClIAl-. IV.] RATIONAL FRACTIONS. 53 ^ J x* + x- + l ^ ^X"+X + \ ^ ^ J {x-\y(.r- + \r 4(.r-]) - -^ + i taii-^i- - ,} + i log(.r= + 1 ) . 4(.r- + l) (10) ri:!^^_j_iog^-W2 + i . ' .1-^ + 1 4 V2 jc2 _|. ^ 7-2 + 1 4- — ^ [tan-' (x V2 + 1 ) + tan"' (x V2 - 1 ) ] . 2 V2 (11 ^ J :r^ + 1 4 a;V2^\ V"2 '"" ar _ ic V2 + 1 ' 2 V2 "*" Vl - ^J' dx 1 , .T2-}-a;V2 + l , 1 log tan" 64 INTEGRAL CALCULUS. [Art. 64. CHAPTER V. REDUCTION FORMULAS. 64. The method given in the last chapter for the integi-ation of rational fractions is open to the practical objection that it is often exceedingly laborious. In many cases much of the labor can be saved by making the required integration depend upon the integration of a simpler form. This is usually done by the aid of what is called a reduction formula. Let the function to be integrated be of the form a;"'"^(a+6x")p, where m, n, and p may be positive or negative. If they are in- tegers, the function in question is either an algebraic polynomial or a rational fraction; if they are fractions, the expression is irrational. The formulas we shall obtain will appl}- to either case. Denote a + 6.r" by z ; then we want | x^-'z^da;. Let ' z^=u and K""' dx = dv, and integrate by p>arts. du = lyz^-^ dz = bnpx"-^ z^"' dx, ar m J m m J This formula makes our integral deix'nd upon tlie integral of an expression like the given one, except that the exponent of a; has been increased while tliat of z has been decreased. We get from [1], by transposition, fx- + "- ' ^r-i rfj; = i^'' _ JL A.™- 1 2P ax. J bnp blip.' Cn.vp. v.] REDUCTION FORMULAS. 56 Change 7?i + n into vi and p — 1 into j), whence m is changed into m — n and 2^ into }) -\-l, and we get J' m-i D J a;"'""^^"*"^ m — u C „, „ , „,, , _ _ bn{p+\) hn{p + \)J L-J a formula that lowers the exponent of x while it raises that of >. Since z = a + tx", 2^ = 2P-i(rf + 6.C''), hence therefore, by [1], in m J J J Cx-^z^-^dx = ^- Hl^i±J!Pl Cx-—^z^-hlx. J am am J Change p into p +1. C:c-^,.dx = ^^:i^ - ^(^^ + ^>P + -) fx-^-^z^dx. [3] J am «?'i "^ Change m into m — n, and transpose. f^..-. ,.,?., = x^-'z"-^ _ «(m-n) p.-„-,,.,,,. ^43 We have seen that Cx"'-'^z''dx = a Cx'^-^zP-^dx + & Cx"' + "-^z'-'^dx, and, from [1], J np npJ 56 INTEGRAL CALCULUS. [AUT. G5. hence Cx''-^z''dx = a ( x"'-^ z''-^ dx + ^^ - — Cx'^-^z''dx, J J np njiJ fx'"-h>'dx= •^'"'~'' + "^'^' i'x"'-'zP-'dx. [;■)] Change p into ;> + 1. and transpose. an{p + 1) a»(j)4-l) -/ Formula [3] enables us to raise, and Ibrniula [4] to lower, the exponent of x by n without affecting the exponent of z ; while formula [o] enables us to lower, and formula [G] to raise, the exponent of z by unity without affecting the exponent of x. Formulas [1] and [3] cannot be used when ?/i = ; formulas [2] and [6] cannot be used when p= —\\ formulas [4] and [5] cannot be used when m — —np ; for in all these cases infinite values will ho brought into the sec- ond member of the formula. G.'>. If // = 1, z = u -{-hx, and our last four reduction formulas become I .X-"- ' z^dx = 5^ — ^ ' I xTzHlx. \:\ J am am J •- -^ J a(p+\) a{p + \) J •- J If »j and J) are integers, and m>0 and ^>>0, a repeated use of [.")] will reduce p to zero, and we shall have to find merely the Tr-'f/x. Chap. V.] REDUCTION FORMULAS. 57 If //(<0 and 2)>0, [o] will eiuible us to raise in to 0, and tluii [5] will enable us to lower 2> to 0, and we shall need oiilv J 'dx X ll'»i>0 and 7)<0, [0] will raise j) to —1, and [A] will then lower m to 1, and we shall need I — . If /H<0 and2><0, [G] will raise 7; to —1, and [3j wui raise //( to 0, and we shall need | — • J xz af'-' dx f J z J a -\- bx b rcjx^r dx ^_ii^i± J xz Jx{a+bx) a " x Hence, when 7* = 1, and ?h and j> are integers, our reduction for- mulas always lead to the desired result. Examples. J x\<(+bx)~ a'^^ X (I'x ^ir'x" Sirx'' A ax* (2) Consider the case where u = 2, rewriting the reduction formulas to suit the case, and giving an exhaustive investi- gation. ^ ^ J {a + bx-y ^ ~ Abia+bx")- 8nb{a-\-bxr) _^_L_tan->.rJ^. H{ab)l \x= "v<"+'->;" . (3) J[.tV(-^- + ")+V(-»- + ")]''x 2n + \ n-\-\ ^^VV -r ; Chap. VI.] IRRATIONAL FORMS. 69 67. A case not unlike the last is (/(.(•, Vc + Vu +bx)dx. Let z= -Vc-h "Va + bx ; z" = c + \ a -j- bx, (z"— c)"' = a -hbx, b Hence b Cf{x. Vf4- y/aTTx)dx (1) Find C (2) Find f- Examples. xdx Vc -f Va + bx dx VI +Vl -.r 68. If the expression under the radical is of a higher degree than the first the function cannot in general be rationalized. The most important exceptional case is where the function to be integrated is irrational by reason of containing the square nwt of a quantity of the second degree. Required | /(a-, Vo -\-bx-\- rx^)dx. First Method. Let c be positive ; take out Vc as a factor, and the radical may be written V^l -\- Bx -\- xr. Let VA + Bx + .-i^ = x + z, A + Bx-hx- = x- + 2 xz + z^, B-2z dx ■■ {B-2zy 60 INTEGRAL CALCULUS. [Art. 68. •>{z-- nz + A)dz -2zy ' and the substitution of these vahies will render the given func- tion rational. Second Method. Let c bo positive ; take out Vc as a factor, and, as before, the radical may be written V-4 + Bx-{- x^. Let ^/A+Bx-\- x^=y/A + xz; A +Bx + :>? = A + 2^A.xz + x'z'', 2 J A .z-B X = —^ 7, ' 1 -z- ^ 2i^A.z^-Bz+^A)dz (l-z^y and the substitution of these values will render the given func- tion rational. If c is n(»£rative the radical can be reduced to the form V^-1 4- Bx — .»-'. and the method just given will present no difficulty. Third Method. Let c be positive ; the i-adical will reduce to -s/A + Bx + a^. Resolve the quantity under the radical into the product of two liinomial factors (x — a) (x — ^) , a and fi being the roots of the equation A + Bx -\-x^ = 0. Let V(.T-a)(.r-/?) = (x-a)z ; (X - a) (X -(3) = {X - a)-2-, {i-zr ^(x-u)ix-(i) = (X- - ,.)z = 1^5^, TiiAP. VI. J IRRATIONAL FORMS. 61 and tlio substitution of those values will make the given funetion rational. If c is negative the radical will reduce to V .1 + Bx — xr, and maybe written y/ {a — x) {x — /S) where a and (3 are the roots of ar — Bx — A = 0, and the metiiod just explained will applv. In general, that one of the three methods is preferable wliic-h will avoid Introducing imaginary constants ; the first, if c > ; a a the second, if c < and > ; the third, if c < and — < 0. — c — a If the roots a and fi are imaginary, and A = ^;j- is negative, it will be impossible to avoid imaginaries, for in that case A + Bx — x^ will be negative for all real values of x. 69. Let us compare the working of the three methods just given b\- applving them in turn to the example j '■ J V 2 + 3 .1- + ar' 1st. Let V2 + 3.v-}-ar = .r + 2; r dx ^ r 2{z--3z + 2)dz 3-2z ^ r 2dz J V2 + 3a; + .x-^~-' (3-2^)^' ';^-3z + 2 J 3-2z = -log(3-22). /: dx = log; log(3 4- 2 a; - 2 V2 + 3 x + or) V2 + 3a; + .'" 1 + 2.r-2V2+3.x' + _ 3 + 2 X + 2 V2 + 3 X + or = log [3 -f 2 X + 2 V2 -f 3 .r + .r-] . 2d. Let \/2-\-3x + jr = ^2+xz; r dx _ r(V2.2--3z+ V2 )rf2 1 -z- JV2T3^+T- J (1-^r '^2.^-:iz + ^2 = 2r_^ = logL±£. (Art.;-i3) J I — Z^ 1—2 62 dx + -dx + INTEGRAL .r a; + V^ CALCULUS. [Art. 65 J V2" + V2 + ; - V2 + ; Sx-\-x' yl2 + -6x + ar^ + 2 + 3a 2 rr'+ 2 ^-1 .X + 2 - 2 - 'dx - :)!? = log 3 + 2.1- + 2 V2 4-3a; + ^ 2V2-3 = log(3 + 2x+2V2+3a;+ar^) - log(2 V2-3) or, dropping the constant log(2^2 — 3), r dx '^ V2 -f- 3 X + X' log(3 -f- 2 X + 2 V2 + 3 .1- + or') , (2) 3d. Let V2 + 3a; + or' = V(a;+ 1) {x + 2) = {x + 1)2; f—^==2f-^^^'—^ = 2C^^=log'-±- JV2H-3X + .X- J (1-22)2 _2 J\-z' 1- /: dx V2 + 3a; + ar' log X + 2 \ .r + 1 = log' + 1 + 2 V2 + 3a; -f af^ + .-K + 2 a; + 1 — .T - 2 = log (3 + 2x + 2 V2 + 3 .r + ar) + log ( - 1 ) , or, dropping the imaginary constant log ( — 1 ) , f-^ ^^ --:= log(3 + 2.r + 2V2 + 3a; + ar'). (3) ^ V2 + 3a;-f- a;- EXAMPLES. O) r ^^L-^,=, = J-iog^''i+^iL=LV|^. ^ ^ J (2 + 3x)V4-x- 4V2 V4 4-2X-+ S/2-a; (2) f-.^ = log(A + a- + V?T^). ./ Vor + x (3) f , ^ r = —\og(— + x^c-\-yfa + bx + cA. CiiAP. VI.] IRRATIONAL FORMS. 63 70. If the function is irriitional through the presence, under tlie radical sign, of a fraction whose numerator and denominator are of the first degi-ee, it can always be rationalized. Required j>(.,^^)... Let ._;-i"-+& " lax -j- m ^„ _ ax + b Ix + m' & — mz^ Iz" — a^ J _ n(am — bl)z"-^dz X: {Iz'^-aY and the substitution of these values will make the given function rational. Example. f dx s/T^ = _ # 3 1/1 --^-y J {\+xy\\+x ^\\\+x)' 71. If the function to be integrated is of the formrc" "'(a+&x")'', m, 71, and p being any numbers positive or negative, and one at least of them being fractional, the reduction formulas of Art. 64 will often lead to the desired integral. Examples. (1) f ^^^^-^ =§sin-'x-^^'-^(:{4-2.r^). (-2) f-A'_^^iog^--^^^-^^ng. (3) f ^^^^ =-(2ax-.x-^)^(^^>^V3«'sin-\f^. ^ .) {2ax-x')\ ^ ' {-2^ 2 J \2u 64 INTEGRAL CALCULUS. [Aur. 72. 72. We have said that when an irrational function contains a quantity of a higher degree than the second, under tiie square-root sign, it cannot ordiudrihi Ije integrated. It would be more cor- rect to say that its integral cannot ordinaril}- be finitely expressed in terms of the functions with whicli we are familiar. The integi-als of a large class of such irrational expressions have been specially studied under the name of Elliptic Integrals. They liave |)eculiar i)roperties, and can l)e expressed in terms of ordinary functions only l)y the aid of intinite series. Chap. VII.] TRANSCENDENTAL FUNCTIONS. 05 CHAPTER VTT. TRANSCENDENTAL FUNCTIONS. 73. In dealing with the integration of transcendental functions the method of integration b>/ parts is generalh" the most etlective. For example. Required | x{logx)-clx. Let u = {logxy, dv = x.dx ; "Iloax.dx dn = ^ . X' XT fx(\ogxy- = -til^K^ -Cxlogx.dx = |[(log.r)'- log.r + i]. Again. Required i e^s'mx.dx. u = sina.% dv = e' dx ; du = co%x.dx^ V = f^, I ^'sin x.dx = f^^sin x — | ^''cosr.rfx, I e'cos.T.'^/.r = r''cos.r + | '■'sin x.dx ; , r r 7 <''(sin.r — cos.c) whence I ('"^sin x.dx = -^ ^ and I e' cos .r . dx = — ^ ^ • 66 INTECJKAL CALCULUS. [Art. 74. _^ 6l0g.T {m-t\y- {m+\y ^ \) {\-xf 1 -X 74. The method ^f' iiitegration h}- parts gfyes us important reduction formulas for transcendental functions. Let us con- sider I sin"x.dx. 11 = sin""^a;, dv = sinx'.cZa; ; dw = (?4 — 1 )sin""^T costc.cZa;, v = — cosa; ; j sin"x.dx = — sin""'.c cos a- -f- (?i — 1 ) I sin"--.r cos^rc.do; = — sin"''xcos;<; + (?i — 1) I (sin^-'a; — sin"a;)dr ; fsin"a*.fZa: = sin""'.r cosx -\ — '^— | sin" "^x. da;. [1] n n J Transposing, and changing 7i into n + 2, we get rs\n"x.dx= sin"+*acosa;-|-^^-t- | sin" + ^x.fZx. [2] 71+1 ?i+lJ -^ In like manner we get / cos" a;.da;= -sin a: cos""' a: -\ ^^^— | cos"~^x'.da;, [3] n 71 J /cos"a;.dx = sina;cos" + 'x + ^?-t^ | cos" + 2x.da;. [4] 71+1 ?( + !.' If n is a positive integer, formulas [1] and [;^] will enable us to reduce the ex[)onent of the sine or cosine tu one or to zero, (^llAl-. VII.] TRANSCENDENTAL FTNCTIONS. G7 and then wo can integrate by inspection. If u is a negative integer, fornuilas [2] and [4] will enable us to raise the ex- ponent to zero or to minus one. In the latter case we shall iK-ed , or I -r^—, which have been found in Art. 51 (<■). cos a; J siua; Examples. / 1 X r • A 7 sin X cos .r / . , , 3\ , 3 (1 ) I snr x.clx = ( snr x +-\-}--x. / ->\ r r, 7 sinxcos'x/ o , r)\ , 5 , . , , (2) I cos x.clx = ( cos-.r + - H ; (snio: cosa;4- .r). J 6 \ AJ IG J sur.^• 2 sin- a; 2 (4) Obtain the formulas /sinh''a'.f?.r=-siuh" ^x'cosh.r — ^^^^^ | sinh"~'.i'.f/x. n n J /sinh".r.f?.i'= sinh"^^^^•coshx•— " j sinlr'+-.i-.d.r, n + \ n-\-\J /eosh".)'.cZ.r= -sinhx'cosh" 'a;-| — ^^^ j cosh" -x.dx. /cosh."x.dx= sinh.7;cosh"'^^a;4- ^ I cosh"^- x.dx. ?i + l n-\-\J ,.. C dx , cosh.1' ,, cosIki;— 1 (^y) I — = — -i- ilosj J suih'a; sinlr^i; "cosh.i-+l 75. The (sin~'x)"f7.T can l)e integrated by the aid of a reduc- tion formula. Let z = sin"^a;; then a; = sin 2, dx = cosz.r/z, and j (siu^' a;) %/.<•= iz"cosz.dz. 68 INTEGRAL CALCULUS. [Art. 76. Let ti = 2", dv = coii z.dz ; da — )tz"~^ dZy V = sin z ; I z"coHz.dz = z"s'u)z — )i I z'"'^ sin z.dz. i z"~''s\uz.dz can be reduced in the same way, and is equal to — 2"~'cos- -(-(n — 1) I z^^'-iiosz.dz; hence i z" cos z.dz = z"sinz -{■ 11 z"~^cosz — n{)i — I) i z"'^ cos z.dz, [1] or I (sin"'.i-)"(/x- = .r(sin~'a-)"-f- xVl — ^^^(sin"'^)""* -)>{)! -\) r(sin-'.r)"--fa-. [2] If >t is a positive integer, this will enable us to make our re- quired integral dei)end upon | dx or j sin"'x.(/j:, the latter of which forms has been found in (I. Art. 81). Examples. (1) Obtain a Ibrmula for I ( vers "^r)" dr. (2) r(sin-'.r)^/a; = .r[(sin-'.0'- ^ • 3 .(sin-'.r)-+4 .3.2.1] -I- 4 Vl -ar'sin-'.r[(sin-'a-)""'- 3 • 2]. 7(). Integration In' substitution is .sometimes a valuable method in dealing with transcendental forms, and in the case of the trigo- nometric functions often enal)les us to reduce the given form to an algebraic one. Let it l)e recpiired to tiud | (/ oina) cosx.fZa;. Let 2 = sin.c, dz = cosx.dx ; I ( /sin.r) eos.r dx — i fz.dz Chap. VII.] TRANSCENDENTAL FUNCTIONS. HO lu the same way we see tliat J (/cosa;)sina;.(fa; =-ifz.dz if z=cosj;, and J [/(sin a-, cos.r)]cos.T. r/.r= j [f(z, Vl— 2-)]r/:r if z = s\n.r, j [/(cos.r, siu.r)] sin.i'.(/.r = — | [f{z, y/\—z-)']dz if 2 = cos J, or. 111010 generally, rf{s\nx, cosx)dx= ( /(2. Vl— 2-) — -^— if 2 = sin. c, -^ Vl-2- f /(eosx, sin.r) (/.r = - Cf{z, VI-2-) — ^^ if 2 = cosx, J J ■ Vl -z^ Since any trigonometric fnnctiou of x may be expivsseil in terms of sin a; and cos a,-, the fornuilas just given enable us to make the integration of any trigonometric function depend on the integration of an algebraic function, which, however, is frequently complicated by the presence of the radical Vl— z^. 77. A better substitution than that of the last article, when the form to be treated does not contain s'xux or cos a; as a factor, . X IS 2 = tan-- 2 'II- • ^ 2r?2 1 his gives us ax = -, 1 +2^ 2z sin X = ■ , 1 -+- 2- 1 -2- cosa-= • : 1+2^' whence J/(sin x, cos x) dx = '2^^f(-=^, T^y 7+?' '" ^ ^ As an examjile, let us find | '- J a + b cos X 70 INTKGUAL CALCULUS. [Akt. 78. Here we have r dx ^ 2 r (h ^2 C '^^ J a + hcoBx J (1 t ;,^)ra I h^-\\ J a-\-h + {a-b)z' a — hJ a + h ^2 \lo? — b^ K^a + h J a—h by I. Art. 77, Ex. 1. Hence f '^ = , ^ tap-^f^/^^- t.^n'^^'Y if a -> ft. ./a + ftcos.'c Va^ — ft- V\a + ft 2/ 78. I sin"*xcos".r.f?a; can be readily found by the method of Art, 76 if m and n are positive integers, and if either of them is odd. Let n be odd, then cos" a; = cos" ~^ a.* cos .r = (1 — sin^a;)~5~cosa;, I sin"'xc^s".T.f?.r = I sin"" it" (1 — siu^a;)~^cosa;.da;. Let 2 = sin a;, (h = cosiK.dr, I sin" a; coB'^x.dx = j 2;"' (1 — z^)~^dz^ which can be expanded into an algebraic polynomial and inte- grated directly. If ??i and n are positive integers, and are both even, I sin"a; cos"a;.da; = l sin".!- (1 — ^'nr xy^dx. sin"".!' (1 — sin-.c)2 can be expanded and thus integrated by Art. 74 [1]. If m or 11 is negative, and odd, we can write cos" .T = cos" ' X cos a', or sin™ x = sin'""^^• sin x, and reduce the function to be integrated to a rational fraction by the substitution of 2 = cos.r, or 2 = sin a;. I sin"*.!; cos".c.f/.c can also be treated by the aid of reduction fornuilas easily obtained. CiiAP. VI I.] TRANSCENDENTAL FUNCTIONS. 71 79. I tau"xdx aud I — — can be handled by the methotls of Art. 78, but they can be simplified greatly by a reduction fonnula. We have I tau"a;.f/.i-= j tan"''-a;tan-.a'.rf.c= j tan" - x {sec' x — 1) clx = I tan" -.rrf(tan.r) — | tan" -x.dXy /tan" 'r /^ tan"x.dx = ~ '- — I tan" ^x.dx; [1] 1 C ^^'^ — fsec^a; — tan-x , , _ rd (tanx) _ r dx J tan".T J tan" a; J tan"x J tan^^V whence ( — '- — = — — | —• [2] J tan"x (?4 — l)tan"^x J tan"^aj Examples. sxv^xcos' x.dx — cos'".x* cos°a; 10 8 , 2sin^x 2sin'x dx = • 3 7 sin^x.fZ^ _ 2cos'ig 2cos^a;. (1) /si (2) I cos^.x' Vsinx*. .,s r 2 -4 7 sin ic COS x/sin^x sin'aj 1\ , x (4) j cos'=« s,o%.rfx- = ^^—(^-^ _____ -j + _. (.')) I = sec. c + log tan-- J sinxcos^x 2 (C) -^^5 T- = seer - —r^ + - log tan -• (7) f-^ = l_ + _L_-f-logsinx. .' tan*x 4tan*a; 2tan-x 72 INTEGRAL CALCULUS. [Akt. 79. (8)f ^^- =-4=log Ja + bcosx sju^-a^ VM^ - \ -Jh + a + ^h-a . tan^" h — a . tail - 4 + tan - (9) ( ^^-^ =^tan->l J + 4 sin a; 3 \ i. (10) r. l"". > =Tlogsinf-logcos^^ + jlog(3 + 2 ./ 3sina; + siu2a; o 2 2 o ... V /• cos xdx ^ V (5 + o sin.r 4 cosx)^ 9 5 + 4 cosx 27 :tan 'frtan^ (12) fix + 6 sin x + c cos a; Va- — 6^ — c^ tan ^ (a-c')tau';^ + 6 (13) Show that the methods described in Arts. 76-79 apply to the Hyperbolic functions. (14) I = , tan- ' f V tanh-Uf^>a. V a + ^ sinha- + <•( cosh X 2 Vc« - «=> - A^ tan" (c — a) tanh o + ^ Vf- - «- - b'' fy^ 1^ VIII.] DEFINITE INTE(;itALS. 73 CHAPTER VIII. D E F I N I T !•: 1 N T E ( n I A L S. 80. lu I. Art. 183, a dcfiuite integral has boen (Icliiiod as the limit of a sum of infinitesimal terms, and has been proved e(iual to the difference between two values of an ordinary inte<;ral. We are now ready to put our definition into more precise, and at the same time more general, form. If fx is finite, continuous, and siiigle-valued between the values x=a and x= b, and we form the sum (Xi — a) fa + (x'a — a^i) A +(%— X2)fXo-{ h (.r„ ,-.(•„ .,)/.r„ j + (^-.^•„ i)A,-i, where rcj, Xo, ^s'-'^n-i ^re n — 1 successive values of x lying between a and b, the limit approached by this sum as n is in- definitely increased, while at the same time each of the increments (Xi — a), (Xo — Xj), etc., is made to ai)proach zero, is the definite integral of fx from a to b, and will be denoted by I f.r.dv. If we construct the curve y=fx in rectangular co-ordinates, this definition clearly requires us to break u[) the projt'ctiun on the axis of X of the portion of the curve between the points .1 and B into n intervals, to multi- ply each interval by the ordinate at its beginning, and to take the limit of the sum of these products as each interval is indefinitely decreased ; that is, the limit of the sum of the small rectangles in the figure, and this is easily proved to be the area ABAiBi- Now the area ABAiB,. found l)y the method of I. Chap. V., i' A r- J U 'U «i a a , r-. r 11 ) ■ [/■^-"■■■].=. -[>•"■■■].■•• 74 INTEGRAL CALCULUS. [Aur. 8L Therefore r/:«-'^-^' = T f/'-''"-^-'' 1 - |!/a^.rf.r . [1] That is, I fx.dx is the increnieut produced in j fx.dx by changing x from a to b. It is to be noted that the successive increments (a^i — a), {x.i — Xi), (0^3 — a'a), etc., that is, the successive values of dx, are not necessarily equal ; and also, that if we multiply each interval, not by the ordinate at its beginning, but by an ordinate erected at any point of its length, the limit of our sum will be unaltered, {v. I. Arts. 161, 149.) 81. It is instructive to find a few definite integrals by actu- ally performing the summation suggested in the definition (Art. 80), and then finding the limit of the sum. (a) I x.dx. Let us divide the interval from a to b into n equal parts, and call each of them dx. Then ndx =b — a. Our sum is S = adx + (a+dx) dx +(« + 2 dx) dx + . . . + (a + (n -l)dx) dx = nadx + (1 + 2 + 3 H \-{n-l))dx^ since ndx = b — a, and the sum of the arithmetical progression H_. + ;3 + ...4-(n-l)=^^i^. 2 ^ ^ 2 2 Hence 5 = — : ^^ — : — -dx. Cirvr. VIII.] DEFINITK INTEGRALS. 75 As we increase n indefinitely, dx approaches zero, and J'* , _ limit ni- — a- (b — a) (Ix ~\ _ b- ir . ^-^^-dr-oL 2 2 J"7~2' (b) Ce^lx. Let dx = n 8 = 6" dx + €"+''' dx + 6"+'^ dx -\ h 6"+'" ' "^ dx = e" dx [ 1 4- e*^ + e'-'^' + e^ H f- e<" '''^J ; but 1 + e"^ 4- e-''^ + • • • + e*" ^^"^ is a geometrical progression, and its sum is g»d» _ I ^ gh-a _ I e*""-! „ 7 /„. ..X f?-P Hence *S = -^ • e" dx = (e" - e") and J^ 6=" dx = (e^ - e") ^^^^ ^ oL^^T^tJ ' dx but as dx approaches zero, -^ — approaches the indeleiininato form - ; but since the true value of r e'da; = e'' — e". (c) I cos'^i s^x.dx. Let dx = -, and let n be an odd number. n Then S = dx + cos' dx • dx + cos'' 2 dx • dx H h cos'' ( n - 2 ) dx • dx + :os'' (n — l) dx • dx = dx + cos^'dx • dx + cos"' 2 dx • dx -| h cos'' (tt - 2 dx) • dx + cos'' (tt — dx) • dx = dx + cos" dx • dx + cos'' 2 dx • dx H cos" 2 dx • dx — cos" dx-dx. 76 INTKCtKAL calculus. [Art. 8L since cos (w — ) = — cos (}>. Hence the terms cancel in pairs, and we have left S = dx and JJcos'x.dx = J^^^ \^b^ = 0. (d) I sin- a;. da;. Let d.r = — , and let n be an odd number. 2)1 S = sin'O- (?.«'+sin^d.v- (?.r+sin^2da;-(Z.r -\ \- s'lu- {71 — 2) dx-dx -f-sin^(?i — l)dx-dx = shv'dx-dx + sin^ 2dx-dx-\ f- sin-| - — 2dx Vz.r + siii^[ - — dxpx = sin^dic-c?x*-f sin- 2dx-dx-\ |-cos-2f/.C'f/.r+cos-rfa;-fte, since sin | ^ — <^ j = cos <^. Then S = dx + dx -j-dx--- = ^^^ dx, since sin^ ^ + cos- <^ = 1 . Therefore S =- -, 4 2 and I sin-a;.dx = ^. (^> i T Here it is best to divide the interval between o and b into unequal parts. Let the values a;,, x.,, a*., ••• .t„_, be such as to form with a and b a geometrical propfression. For this purjjose take 7 = \l , so lliat a< b. Crap. VIII.] DEFINITE INTEGRALS. 77 Then the values in question are a^, aq^, aq^-'-aq" ', and the intervals are a{q — 1), aq{q — 1), aq'^ {q — I) ••• aq" ' ( 7 — 1 ) , and the sum ^^ a(7-l) ^ aq{q~l) ^ (Uf {g - I) ^ ^ aq" ' {q - I) a »(ry-l). aq aq" aq" » To prove our division legitimate we have only to show tliat each of our intervals, a(g — 1), aq{q—l) ••• o(y"-'(7.— 1 ), approaches the limit zero as n increases indefinitely. Since ,. h '^ =a the limiting value of q as )t increases must be 1, as otherwise ^'™'^ (/" would not be finite. Therefore j^'^'^ [aq^q - 1 )J = ^^f^ [aqHq - 1 )] = 0. We have then X ''dx limit rcrn limit r / im limitr / .x-. I log- limit a 7 = 1 Ih^/ (7-1) n log q = log limit 7 = 1 log logr/^ ;] ^*7 = lLlogyJ -a For r^i =rii=i. = log/y — log a. a; 78 LNTEGKAL CALCULUS. [akt. «2. Examples. (1) Prove by the methods of this firticle that jy log a (2) By the aid of the trigonometric formuhis co.s^ + oos2^ + cos3^H |-cos(;/ - 1)^ = I sin nO etn 1 — eosji^ L siu^ +siu2^ +sin3^ H h sin(ji- 1)0 = ^ (1 — cos 71^) etn sin?(6 L prove that | cos.x'.d.x = s'mb —sin a, and I sina'.cZa; = cos a — cos 6. (3) Show that p'siu^r.c/x- = 0, and that | cos-a;.(?.x' = -• ^0 2 x'"dx = — , usinsj the method of »i + 1 Art. 81 (e). 82. "When the indefinite integral can be fonnd, the definite integial | fx.dx can usually be most easily obtained by era- ploying the formula [1] Art. 80, and this can always be done with safety when fx is finite, continuous, and single-valued between x = a and x = b. Of course, if the indefinite integral is a multiple-valued func- tion, we must choose the values of the indefinite integral cor- responding to X = (I and x = b, so tliat they may be ordiuates of the same branch of the curve ?/ = i fx.dx. Chap. VIII. 1 DEFINITE INTEGRALS. 79 Consider, for example, | -r-^^.- The indefinite integral C dr »/-i 1 +a- I — ^, = tan ^x and tan 'a; is a multiple-vahied function. Indeed, y = tan Kv is a curve consisting of an infinite number of separate branches so related that ordiuates corresponding to the same value of x differ by multiples of tt. On the branch which passes through the origin, when x= — \, .v=?tan 'x=— ''; 4 on the same branch, when x=l, ?/ = tan \v=-- On the next o i branch above, when x = —l, y=tan^x = —; and when x= 1, y = — • On any branch, when x = — 1 , y= tan ^x = — - + mr; 4 4 and on the same branch, when x= 1, y = - + »7r. Hence f -^, = tan '(1) - tan-H-l) = ^ + ^ = 3.- Jil + .r 4 4 2 X^ dx _OTr Stt il+a^" 4 4 = - -\- nir — I \- 7117 = -• il+ar 4^ ^ 4^ y 2 By I fx.dx we mean the limit approached by | fx.dx as i is indefinitely increased. Examples. (1) Work the examples of Art. 81 by the method of Art 82 jr ^is'mx.dx StT TT ~2' ^H' = V2-1. cos-x c/u cr + ar 2 a ''^ =iVa(V5-l). 80 INTEGRAL CALCULUS. [Art. 83 (o) C J"^-'' ^E if „ > 0. .„„l _ '^ if a < 0, and if ,i = 0. J» a' ■\- X- •> 2 (6) f e "rfx = ^ if a>0. Jo (I e'" siii?Ha;.da; = , if a>0. (8) I e~"^cosmx.dx = ^r-^ — r, if a > 0. a- + ?u^ 2 a; cos <^ + ar 2 sin <^ (10) p ^ ^■. Jo 1 + 2 X cos <^ + o;- sin< 83. When fx is finite and single-valued between x = n and x = b. but has a 7t?ii7e discontinuity at some intermediate value x = c j fx.dx = I /.c.(7.c 4- I fx.dx, and tlierefore i fx.dx can be found by Art. 82 when the iudetinite integral i fx.dx can be ()l)tained ; but when fx becomes infinite for x=a, or for x = b, or for some intermediate value x = c, special care must l>e exercised, and some special investigation is usually required. fx.dx approaches a finite limit as c approaches zero, this limit is what we shall mean by X fx.dx; if I fx.dx increases indefinitely as e approaches zero, we shall say that I fx.dx is infinite; and if | fx.dx neither :iii[)io:i(hes a finite limit nor incieases indefinitelv as « Chap. VIII.] DEFINITE INTEGRALS. 81 appronclu's zero, wc shall say that | fx.dx is indeterminate. fx.dx can be safely employed in mathematical work. If J'x is iutiuite when x = b and j fx.dx approaches a finite limit as e approaches zero, that limit is the valne of I fx.dx. If fx is infinite when x = c, and each of the expressions 1 fx.dx and I /.l^f?.^• approaches a finite limit as c approaches zero, the snm of these limits is I fx.dx. Should eitiier or both of the expressions, X fx.dx, I fx.dx, fail to approach a finite limit as e approaches zero, I fx.dx is either infinite or indeterminate, and cannot be safely used. When the indefinite integral of fx.dx can be obtained there is little ditliculty in deciding on the nature of I fx.dx in any of the cases just considered, or in getting its value when that value is finite and determinate. For example, Ji'^dv \ — is infinite, since U X I — = log X and | — = log ( 1 ) — log e = log -, J X ° J, X ^^ ' € and increases indefinitely as e appinaches zero. dx <^> i r / not finite and determinate, for .T* dx _^j^^^^.l+a; 1 — a;- " ' \ —X Jo ] -a- - ■ \ e J and increases indefinitely as e approiu-lies zero. 82 INTEGRAL CALCULUS. [Art. 84. (c) I - is finite and determiuate, for / Jo ^Hfi sld'- x^ dx Va^- ar ' ' dx . iX sin"^ '-^^ — - — sin~^ y/aF-x" « " and its limiting value as c approaches zero is sin-'(l) or |- — ^ r is finite and determinate, for (l-x-)» Jo (Tir^i-^' ^' ^^^' and its limiting value as e approaches zero is f — |. and its limiting value as c approaches zero is — | — f , and consequently Ji (l-x)^ - ■' "^ '^ 84. When, as is sometimes the case, the indefinite integral cannot be obtained and the function to be integrated becomes infinite at or between the limits of integration, we have recourse to a very simple test which is easily obtained by the aid of the following important theorem, known as the Maxinmm-Minimum Theorem. If a given function of x is the product of ttvo functions both finite, continuous, and single-valued, one of which (x) does not change its sign between x := a and x ^ b, and if M is algebra- Chap. Vlll.] DEFINITE INTEGKALS. 83 ically the greatest and m the least value of the other factor f(x) between x = a and x = b, J f (x) <^ (x) dx lies between M I <^(x)dx and m ( <^(x)dx. To prove this theorem, let us first suppose tliat «^(.r) is positive between x=^a and x = b. M — f(x) is positive for the values of x in question, [M — /(^)] (a-) is positive, and, therefore, f\M-f(x)](.r)dx>0 and ^^Sy (•'■) '^■'' > X'-'"'^'''^ '^ '^■'■^ '^■^* ^^^ /(a-) — ??i is positive for all values of x between x = a and x = b, [f(x) — ?n] (^)dx>0 and C f(x) (x) dx > m C cj> (x) dx. (2) f(x) (f> (x) dx lies between M I 4> (x) dx and 7n I <^ (x) dx. It is easy to modify this proof to meet the case where <^ (x) is negative. We can briefly formulate the result of the Maximum-Mini- mum Theorem as follows : fy(x) (x) dx =f($) jy (x) dx, (3) where i is some value of x between a and b. Let us apply this theorem to the consideration of I f(x)dx when/(a;) becomes infinite for x^a. 84 INTEGRAL CALCULUS. [Art. 84. lu order tliat ,' 1 ./'(•'') ff^ should be finite and de- terminate it is easily seen to be necessary and sufficient that limit r limit / C'^lw n / M u i -. . II JK-'')'!^'] should be equal to zero. Let us write /(.r) in the form (■^•~ ^0\/'(a-) ^^^^ ^^.^ < /: < 1. ^ ^ (x — ay -7 is positive for all values of x greater than a. (x — up =tf-«)'/(f)£:-^. cl-* 1-t - limit , limit £j(^) d-r'j = a - ay-fd) ^. ; « < ^ < a + c ; does not increase indefinitely as $ approaches a. Calll^ — dtri, whence $ = a -\- rj. Then a sufficient condition that I fix) dx shall be finite and determinate when f(a) = oc is that Tff{(i + rf) shall not increase indefinitely as rj ap- proaches zero, < A- < 1. If we write f(x) = ^^ )J \ ) and proceed as above, we can sliow that a necessary condition that I f{x) dx shall be finite and determinate when f(a) = oo . limit r ^/ I NT A ClIAP. VIII.] DEFINITK INTKCIIALS. 85 If /(/>) = GO our sufficient condition is tliat rf'/ij) — -q) shall not increase indefinitely as 17 d= 0, < A- < 1 ; and it. /'('') = <» that neither ri\f{c — rj) nor rff{c -\- rj) shall increase indefi- nitely as 77 = 0, < ^- < 1. Let us apply our tests to the examples considered in Art. 83. rUlx . limit Ft?"! , (d) I — =:x because , - =1. X* ff.r -^ is indeterminate, for t r V ~\ _ ^i"iit r_i "I _ limit and lim V mit r T] ~\ limit F — 1 ~| _ _ . = Oil - {rj+ If j- y .^Ol'l^y]- *• J"" dx — ' is finite and determinate, for \'^ :Oif^ 1, it is infinite. is, then, finite and determinate if m <, 1, but infinite if m = 1 or ?>i > 1 ; and we reach the result that r('»4)' dx is finite and determinate if 7i>— 1, but infinite if ?i = — 1 or ?i<— 1. Examples. (1) Prove that C}^.cU. fi^.do., r^iogfi±^\ Jo \-x ' Jo l-or' ' Jo a; \\-x)' are finite and determinate. (2) Prove that J^^ , C.^1^1 , where m and ?i are inte*■•! that l)v | fx.dx we mean the fx.dx as b is indefinitely increased, and, /^ as we have seen, if the indefinite integral I fx.dx can be found, there is no difficulty in investigating the nature of | fx.dx and in obtaining its value if it is finite and determinate. There are, however, many exceedingly important definite integrals of the form I fx.dx whose values are obtained by ingenious devices without employing the indefinite integral, and these devices are valid only piovided that the integral in question is finite and (leleruiiniiti', since an infinite value not recoy;nized and treated Cn.vr. Viri.j DEFINITE INTEGRALS. 89 as such, or a value absolutely indeterminate, renders inconclu- sive any piece of mathematical reasoning into which it enters. If we construct the curve y =/r, I Jx.dx is the limit- ing value approached hy the area ABB^A^, as OB^ is in- definitely increased ; and in order that this area should be finite and determinate, it is clearly necessary and sufficic]it that the area BCCiBi should approach zero as its limit as first OCi and then OB^ is indefinitely increased. That is, limit :tC=t[x>'-*])=«- 86. A sufficient condition that limit r limit i ^ oc \JrL{S>y-)\='' can be easily obtained by the aid of the Maxivmm-Mlnimum Theorem (Art. 84). Let /(a-) be single-valued and continuous. We can write /(x) in the form k>\\ then by (3) Art. 84. iiid limit r f V w "1 ->'<'^^) 1 J ^t 1 not increase indefinitely as ^ increases indefinitely. 90 INTEGRAL CALCULUS. [Art. 86. If, then, [icy(a*)]^=^ is not infinite, k>l, I f(x)dx is finite and determinate, (a) As an example of the use of this test we will prove X" e~^^dx finite and determinate. e ^' is single- valued, finite, and continuous for all values of x. limit X x^'e ^° , A; > 1, is easily found and proves to be zero. Hence, I e-'^'dx is finite and determinate. (b) Let us consider f dx. Jo X sin ax . . . IS equal to a when x = 0, and is finite, continuous, and single-valued for all values of x. Let a be a given constant ; then Jf'^sinaa-, fs'max^ , r^^ sin ax, dx= I f/x-j- I dx, ox Jis X Jo. X and I dx is finite and determinate. Jo x By integration by paHs. /sin ax . cos ax If cos ax , dx = I 5— dx, X ax a J x^ X" sin ax , cos aa 1 T* cos ax , dx = I — -— • dx, X aa a J a. x^ , C^ cos ax - . _ . , - ana I — -r-dx is finite and determinate since Ja x^ limit F a;* cos ax ~\ _ limit rcosn'.r~|_ c i ^ , ^ r, a; = oo |_ x^ J a- = oo L ^ J (r) i cos (x^) dx is finite and determinate, for cos (x^ is finite, continuous, and single-valued for all values of x, hence, Chap. VIII.] DEFINITE INTEGHALS. 91 j COS {x^)dx is finite and determinate ; and - r^ sin(.r-)f/.r . „ . and J^ ^^^^ — IS finite and determinate since limit Fx^' sin (.r-) n ^ =0. I ) J 92 INTEGRAL CALCULUS. [Art. 87. For our example this becomes J ^ ' -m+1 -m + lJ When X = 0, aud also wheu x = o, ' ^ '- = 0, Hence f>-i(a2_ x-^r'dx = ^^^""' ~ ^^ fa"" X"' - •^•')~^cZx ; »/o m — 1 »/o J" V(a2 _ a^) -i da: = - . a^ f V (a- - ar )"^ da; 6 »/o = 5.^.^4 rV(a2_.r^)-4daJ 6 4 Jo = 5.§.1.„. f rf-'' 6 4 2 Jo r Therefore Vu- — X"^ ic^dx 1 3 5 Tta^ Va--x- -2 4 6 Examples. x'dx - . - a* Jo V^^S^r^ 3 2) I \!a- — x-'dx =^- c/o 4 3) C^ yf^f^^ . df = 1 . ''^-' Jti 4 4 4) j;v(„=_a-y.*.=i.i 1 -^ 7ra". ^. C'i . „ , \:?>.h ..An — 1) TT , 5) I siu x.ax =— ^- • when ?i IS even ' Jo 2.4.6...?t 2 = ^^ wlien n is odd. 3.5.7... n Chap. VIII.] DEFINITE INTEGRALS. 93 n IT (6) Show that | cos"a'.(Zx= | '^\n"x.dx. r\ a?"dx ^ 1.3. 5. ..(271-1) TT ^'' Jo Vn^' 2. 4.6. ..2)1 " 2* Suggestion : let x = sin ^ ^ ^ Jo VI^T^' 3.5.7. ..(2u + l)" (9) From Exs. 7 and 8 obtain Wallis's formula n 2.2.4.4.6.6.8.8... 2~~ 1.3.3.5.5.7.7.9...' r^ x^"dx r* xr"-^^dx Vl-ar^ Suqqestion : I — =zz= > I , > I —;=■ Jo Vl-ar^ »^o Vl-ar »^« Vl 88. When in finding i fx.dx the method of integration by substitution is used, and y=Fx is introduced in place of x, we can regard the new integral as a definite integral, the limits of integration being Fa and Fb, and thus avoid the labor of re- placing y by its value in terms of x in the result of the indefinite integration. Let us find f e'"Vl - e"''" • dx. Substitute y=e". dy = ae'^dx. Hence C e" Vl — e-" . dx = ^ j V 1 - y- . dy. When x = — 00, ^ = 0, and when x = 0, y=\. Therefore fV'Vl-e^^" . rfx = 1 f Vl-/ • dy = -^. J-oo <^.'o 4 a 94 INTEGRAL CALCULUS. [Art. 88. There is one class of cases where special care is needed in using the method just described. It is when y has a maximum or a minimum value between x = a and x=^b, say for x=:c, and X is consequently a multiple-valued function of ?/. For suppose 1/ a maximum when x = c, then as x increases from a to b, y increases to the value Fc, and then decreases to the value Fb, instead of simply increasing or decreasing from Fa to Fb. If y is a multiple- valued function of y, and it will always happen that when y passes through a maximum, we pass from one set of values of x to another, and therefore from one set of values of y to another, and in that case it is necessary to express our required integral y.dy + I y.dy, taking pains to select the correct set of Fa ^Fc values for y in each integral. If y is a minimum between x = a and x = b, essentially the same reasoning holds good. A couple of examples will make this clearer. (a) Take C x.dx V2 ax — x' Let ?/ = 2 ax - x^. Then ^ = 2 (a - x) = when x = a. dx — ^ = — 2, and y is a maximum when x = a. dx" ^ X = a ± Va- — ?/, dx=:^-^y^- 2V«^ Since -'- is positive from x = to x = a, and negative from dx x=a to a;=2a, dx= -JL= fi»d x = n — yjd- — y from 2 Va^ — y dy x = to x = a, and dx = ■ , and x = a -\- V(r — y 2 Vtt^ — y from x = a to x = 2 a. CiiAi-. VIII.] DEFINITE lNTE(;itALS. 95 Hence J p"' xdx _ r" xdx r -" xdx " V2 ax — ar ^^ V2 ax — ar' •^'' V2 ax — Very — y- '^^''- \ld-y — y^ 2^/0 yja^y^y^ 2»/U yja^y^y^ = r °' ^^^ =7rft. (Ex. 7, Alt. 84) *^°\Jd-y — y- «/o(sina; + cosa;)^ Let V = sin .); -|- cos x. -^ = cos a; — sin .r = wliou x = -• da; 4 ^= - siu.r — cosa; = — V2 when x = -- Tlierefore y has dx' _ ^4 a maxiimim value \/2 wlien x = ^. 4 2/ = sin.r + cosx= V2 . cosf j — x\ y ,7.._^ f^.V cos^-^, dx = ± 4 V2 V2 - f Since ^ = and ^^"-'^ < when x-=-, it follows that -^ is positive from x = to -J-' = p and negative from .v = - to x = • Hence we have » IT » X ^ da; ^ ri__^^i__ 4. r!__J?!L_ (sin.i; + cosa;)2 Jo (siux + cosx)- J„(siua; + cosj;)' 4 J-^v^ dy _ /-' dy _ g f^^ '^■' 96 INTEGRAL CALCULUS. [Art. 80. Let -^ = sin^; V2 IT (siux + cosic/ and u (sm.i- + EXAMPLE. dx = 00. cos .c)- 89. Differentiation of a definite integral. We have seen in Art. 51 that a definite integral is a function of the limits of integration^ and not of the variable with respect to which we integrate ; that is, that | fx.dx is a function of a and Z>, and not a function of x. Strictly speaking, I fxAx is a function of a and &, and of any constants that fx may con^ tain, where by constant we mean any quantity that is indepen- dent of X. If the limits a and b are variables, they are always indepen- dent of the X with respect to which the integration is performed, which must from the nature of the case disappear when the definite integral is formed, as it always ma}' be in theory, from the indefinite integral ; and this assertion holds good even when the same letter which is used for the variable with respect to which the integration is performed appears explicitly in the limits of integration. Thus if we write I sin.r.r?.r, the x in sina;.c?.r and the x which is the upper limit of integration do not represent the same variable, and are entirely unconnected. Indeed, the former x Cn.vi'. VIII. ] DEFINITE INTEGRALS. 07 may be replaced by any other letter without affecting tiie value of the integral. For Xs'inx.dx = 1 — cos X. Let us now consider the possibility of differentiating a definite integral. Required i)„ | f(x, a)dx, where a is independent of x, and a and b do not depend upon a, and D^f{x, a) is a finite continuous function of a for all values of x between a and h. We have n£f(x,a)dx = limit Cf(.l\ a -\- \a)d.r -ffi.-'-^ ")'^-' Aa limit Aa = ^ PY lilllit r f(.r, a + Aa) —/(■»•. a) "[ \ ^^ . Hence, ^-f/i^' '^-^ =X' '^^"'^*^"^' ''^-' ''■^' ^^ -* and we find that we have merely to differentiate uiuUt the sign of integration. 98 INTEGRAL CALCULUS. [Art. 89. If Daf(x, a) becomes infinite for some value of x between a and b, or if one of the limits of integration is infinite, the proof just given ceases to be conclusive and [1] must not be assumed to hold good. The truth of the converse of the proposition formulated in [1] can be easily established by differentiation, and we have J M /(a;, a)t?a; Jc?a or even £'\^£nx,a)dx~jda =£[_£' f(x, a) da^dx, [3] if a, b, c, and d are entirely independent. [2] and [3] are of course subject to limitations easily in- ferred from the limitations on [1], stated above. If, however, in [3] b is infinite, it can be shown by the aid of the Maximum-Minimum Theorem that a sufficient condition that [3] should liold good is that it shall be possible to find a value of X such that for that value and for all greater values x^'f(x, a) shall be less than some fixed value for all values of a between c and d, k being greater than 1. If d and b are both infinite there is also the corresponding condition involving x^f{x, a). We are now able to state a sufficient condition that [1] shall hold when b is infinite. It is that it shall be possible to find a value of x such that for that value and for all greater values x^'Daf{x, a) shall be less than some fixed value. Suppose now that we are dealing with variable limits of integration. CuAr. Vlli.J DEFINITE INTKCIiALS. 99 Let lis find first — I fx.dx. rJzJ.. Let C fx.dx = Fx, then i \fx.dx = Fz - Fa ; r and since bv definition ^" = A', it follows that —=fz. dx " dz Hence ± Cp.,,. = 'll^^^I^=fi. [4] dzJa dz • "- -^ In the same way it may l)e shown that |J>..<.- = -,i. [5] Let us now take the most complicated case, namely, to find d C^ — I /(a;, a) dx, where a and h are functions of u. da J a. Let f/C^, ") d.i; = F(^x, a) ; then u = r/(.r, a) dx = F{b, a) - F{a, a) , da da (Za but as b and a are functions of a, , dF(a,a) »-, 77,, ^ dff , TV 7-./ N and — ^--^— ^ = Z)„i^(a, a) - + />„i=^(a,a), da du by L Art. 200. D,F(b,a)=f{b,a), D„ F{a, a) =f{a, a). Hence '^" = D. lF{b, a) - F(a, a)] +/(6, a) 'f' -./( .«, a)'^, da d,i (/a -f C"f(x, a) dx = r(Z>„/(.r, a)) dx -\-f{b, a) f -/(.(, a) '^^^ [H] aa«/a «/« Ott Utt 100 INTEGRAL CALCULUS. [AliT. 91 Examples. ■» siu (x + ?/) dx = (a; -f- 1 ) sin {xy + y) — ainy. (1) Af (2) — Cx^(1x = - ^ . clxjo 3 (3) ^JVT^ 90. When the indelinite integral cannot be found, the prob- lem of obtaining the value of the definite integral usually be- comes a more or less difficult mathematical puzzle, which can be solved, if solved at all, only by the exercise of great inge- nuity. iSome of the results arrived at, however, are so impor- tant, and some of the devices employed so interesting, that we shall present them briefly here. But we must repeat the warning that most of the methods are valid only in case the definite integral is finite and determinate ; and erroneous results have more than once been obtained and published when a little atten- tion to the precautions described in Articles 83-86 would have prevented the mistake. 91. Integration by development in series. (a) f'i^^.dx. (v. Art. 84, Ex. 1.) Jo I —X ,J_ = (1 - .r)-' = 1 + .'• + .r -f -r' + •.., if .r < 1 . 1 — a; — M — (Jx = j ( log X -\- X log X + .r lou a' + • • • ) dx. I — X Ju Jx"\orrx.dx = -• (v. Art. 55 (a).) \, ^ (» + l)' Therefore Jn 1 -X >^l-^2-' ;5- 4' / 6 (v. Todhuntcr's Trigonometry, Chap. XXIII., Ex. 1.) Chap. VIII.] DEFINITE INTEGRALS. 101 (6) r log/'^^liiV/x". (v. Art. Ht;, Kx. L'.) ^''^ (J^t) = ^*'- (r^') = i^^g ( 1 + « ') - i"s^' ( 1 - ^ ') ~^ ' 2 3 4 ■" V '^ 2 3 4 ■■/ (I. Art. 130.) Hence r-(?^)-=^'re-^^T-9'- dx = 2(1+1 + 1 + 1 + o- o- < - But i, + l+i + i + ...=^ 1- A- :)- f 8 Therefore (v. Tod hun tor's Trio-., (liap. XXIIl., Ex. 1.) + 1 i -<^ -=r (1) 1 1^ ^^^s-^' rfx Examples. 12 (4) 1-3Vm if A-< 1. 102 INTEGRAL CALCULUS. [Art. 92 92. Integration by ingenious devices. (a) C'logs'mx.dx. (v. Art. 84, Ex. 6.) Let u= i logs'mx.dx. Substitute y = - — x. u = — i log cos y.dy = i log cosx.dx. 2 IT n 2 M = I (log sin x + log cos x) dx = | log (sin x cos x) dx s= _ !! log (2) + ( log sin 2a.(Za; 2 Jo «= - 1 log (2) + ^ J"log smx.dx. I log sinx.(/x = j "log sin. 7;. (?.»•+ j logsinx.fia; = it -\- I log sin.x\c?a;. Substitute y = n — x. Phap. VIII.J DEFINITE INTEGRALS. 103 « I log sin x.(fx = — I log sin y.d)/ = i log sin x.dx = u. Hence 2?< = — ^ log(2) + M, and i<= ( log sin a-. (/.r= ) log eosx.(7.f = — - los;(2). [1] c/o »/o " 2 ■" (6) f e-''(f.r. (v. Art. 86 (a).) Let u= \ e "- dx, and let x = u y ; then ^l= i ae-'^-""- dij = \ ae-^'-^'cU*, ?< r"e-«= r7« = u- = j ' Y (*"ae-''+^=''^= da") cZx, by [3], Art. 89. But J) 2 1+ .t-2 Hence %e = - I — ^ — .; = -, 2 Jo 1 + .r- 4 and I e'-dx— Vtt. [2] Jo 2 rsinfflx^^^ rv. Art. HG(^).) ^ ' J\i X AVc have 1 = Ce'^da if .r > 0. (Art. H2, Kx. 6.) a: Jo 104 Hence INTECJRAL CALCULUS. ]dx e~" i,\nnix.(la]dx [Art. 92 clx= \ sinm.f I e-'^Ula ox Ji) \ J» J Jo \Jo = i fi e'"s'mmx.(LAda,hy[3],Avt.H\). = r4^. (Art. 82, Ex. 7.) Jo a- + Vl' Therefore J**sinw X '.dx= - if m>0 = - 1 if m < = if m = EXAMFLKS. [3] by Art. 82, Ex. (1) J a; log sin a;, da; = _ ZL iog(-2). Siifjyestion : let .x- = tan i e "'^'da; Jo X 2a .dx =0 if m<- 1 or m>l = - if til, = — 1 or ?yi = 1 = - if -1 .i-.rf.r. (v. Art. 8(5, Ex. 2.) c/O Let w= I e~"'-'' cosbx.dx^ then Li* = _ I a-e ""'" ^'mhx.dx. db Jo Integrating In' i)arts, I xe "'"^'sinto.f/.r = - I e ,/n 2 r/%/" -'''cos^.i;.»/.i-= —u. 'lor riieretore — = ; «, db 2 a- dn b ,, — = --— rf&. w 2 a^ 108 INTEGRAL CALCULUS. L-^T. 95. lutegraliug, we have 4 a- or xi=Cie *"\ Wlien & = 0, u = Ce "'^'dx = '^^- (Art. 92, Ex. 3.) Jo 2 a Hence u= i e "''^ cos bx.dx = ^^e"*^. [2\ Jo 2a Examples. (1) I ..«.« = tan '— . .'<) X a Uiggestion: —- — - = 2 ( 1 + ar Jo 95. Introduction of ivuiginary constants. C COS {af)dx. (v. Art. 86(c).) We have f e "'^'' dx = — \^. CArt. 92, Ex. 3.) J.I 2 a JjCt a^ = c- V — 1 = '•- [ cos - + \'— 1 sin - Tiien a = r/'cos'' -f- V^ sin-") = ^-- ( 1 + V^), (Art. 2.^.) and L = ___l^^_= 1 (1_V=T). -^« CV2(1 + V-1) 2cV2 2 SdQoestion : — ^ — - = 2 I oe "'^'"^''''(/a. Chap. VHI.] DEFINITK INTEGRALS. 109 %Jo 2c \ 2 But e ''^^^^ ' = cos (r-'or') - V^ sin (rar') . ([;-i]Art.:n.) Therefore ( cos (c-.r ) dx — V^ f sin (c" .r) c7.f = J- * - ( l _ V^), ./o Jo 20 \2 and f cos (c^a^) dx = C siu (rx-) dx = i- J-. [1] (Art. 17.) Let c = 1 , and r cos (.r^) c?.^; = C siu (.i-) f/.c = ^ J^. [2] If we substitute y = .r in [2], we get r^:mdy =r^,/,=j^. [3] Jo V2/ -^" VZ/ ^l^ Gamma Functions. 96. It was shown in Art. 84 that I [ log - ] dx is finite and determinate for all values of 7i greater than —1, and inlinite when n is equal to or less than —1. The substitution of y = log- reduces this integral to | y"e 'dy, or, what is the same thing, to | a;"e 'dx; and in Art. 8G, Ex. 3, the student has been required to show that this integral is finite and deter- minate for all values of n greater than —1. I X" e ^dxz=z — X" e ' + n i .<•" ' e ^dx, by integration by parts. 110 INTEGRAL CALCULUS. [Art. 9G. If n is greater than zero, X" e'' — when x = 0, and a;"e ' is indeterminate when x= x. Its true value when a;= 00, obtained by the method of I. Art. 141, is, however, zero. Therefore | x"" e~'' dx = n \ x" ^e~'dx [1] for all positive values of n. If n is an integer, a repeated use of [1] gives Jx" e'" dx = nl I e'dx ; but I e~'rf.r = 1, and we have 1 x" e~' dx = n ! [2] provided that n is a ■pusitive tvJiole number. If n is not a positive integer, but is greater than — 1, I x"e~'dx is a finite and determinate function of n, and its value can be computed to any required degree of accuracy by methods which we have not space to consider here. I af~^C'dx is generally represented by r(n), and has been very carefully studied under the name of the Gamma Function. If n is a positive integer, we have from [2] r(» + l) = n!. [3] From [3], r(2) =1. [4] Since r(l) =( x'^e'dx^Ce'dx, r(i) =1. [5] We have always from [1] r(n + l)=»r(70, [6J if n is greater tlian zero. Chap. VIII.J DEFINITE INTEGRALS. Ill Siuce j x"e ^dx is infinite when n is equal to or less than — 1, it follows from the definition of r(n) thtit r(>«)= ^ 'f n is equal to or less than zero. It has, however, been found convenient to adopt formula [6] as the definition of r(7i) when n is equal to or less than zero, and to restrict the original defi- nition to positive values of n. The result easily deduced is that r(?i) is infinite when n is equal to zero or to a nt-gative integer, but is finite and determinate for all other values of u. 97. We may regard the formula r(« + l) = «r(n) as a sort of reduction formula; and since each time we applv it we can raise or lower the value of n by unity, we can obtain any required Gamma Function by the aid of a table containing the values of T (n) corresponding to the values of n between any two arbitrarily chosen consecutive whole numbers. Such tables have been computed, and we give one here con- taining the common logarithms of the values of T{n) fi-om w = 1 to n = 2. Tire table is carried out to four decimal places, and each logarithm is printed with the characteristic 9, which, of course, is ten units too large, the true characteristic b6ing —1. io + iogrr(n). n 1 . 3 * 6 6 7 8 9 1.0 9975 9951 9928 9905 9883 9862 9841 9821 9802 l.l 9.9783 9765 9748 9731 9715 9699 9684 9669 %55 9642 1.2 9.9629 9617 i 9605 9594 ' 95S3 9573 9564 9554 9546 9538 1.3 9.9530 9523 1 9516 9510 9.S05 9500 9495 9491 9487 94S3 1.4 9.9481 9478 9476 9475 9473 9473 9472 9473 9473 9474 1.5 9.9475 9477 9479 9482 9485 W88 9492 94% 9501 9506 1.6 9.9511 9517 9523 9529 9536 9543 9550 9558 95r.6 9575 1.7 9.9584 9593 9603 9613 9623 9633 9644 %56 9667 %79 1.8 9.9691 9704 9717 9730 9743 9757 9771 9786 9800 9815 1.9 9.9831 9846 9862 9878 9895 9912 9929 9946 9964 9982 112 INTEGRAL CALCULUS. [Art. 97. Such a table enables us to compute with Gamma Functions as readily as with Trigonometric Functions, and consequently the problem of obtaining the value of a definite integral is practically solved if the integral in question can be expressed in terms of Gamma Functions. For example, let us consider (a) I a.*"e "■'dx. Let y = ax ; then I x"e "^dx = | y"'e~''dy = | x"e-'dx. Jo a"''c/o a"'^Vo Hence rVe-°'c?a; = ^ ^^ + ^\ [1] c/o a"'*' provided that a is positive and m> — 1. (6) C a™ Aog \Xdx. (v. Art. 84, Ex. 4.) • Let y = — \ogx. then rVAogiy'dx= (' y"e^'^^'^»dy. Hence, by [1], if m>—l and n > — 1 . (c) Ce-^'dx. Let y = x^; then \ e'''-dx=\\ ^dy = \\ x'^e'dx. v/o Jo Vy »^" Hence C e"dx = ^r (^) . [8] Cu.vi'. VIII.] DEFINITE INTKGIJALS. 113 98. r x"' 1(1- .*•)" ' (/.«• = 7i (m, n) [1] is au exceedingly important integral tluit can be expressed in terms of Gamma Functions ; it is known as the Beta Fniicdon, or the First Eulerian Integral, r{n) being sometimes calU-d tin- Seco7id Eulerian Integral. In the Beta Fnnction, m and n are positive, and B{m,u) is always finite and determinate. (v. Art. b4, Ex. o.) In ( x"' ^ {I — x)"hlx let v^l-x-, and we get fx-^-'il-xy 'r/.r= fV '(1 -.'/)'" ''/.V, or B{m,n) ^ B{n,ni). [2] In f a-"' 1 ( I - x) " 1 (/.f let x = ^L^ , and we get (•'.'.-(1 -■>■)■ '.te= C:!-:!*^^ = f ^^_,te. Jo ^ Jo (1 +?/)"•+» Jo (I -|-.r)'"+" . Hence i ^'-' clx = B{m,n). [:?] Jo (I +a;)'"^" *■ ■' We have seen in [1] Art. !»7 {a) that c/o a"*^' Hence V{m)= \ a'"x"*'^e"^dx, T(m)a" 'p ••= i a-+" ia-'"-'e-''<'+''rfx, r(m) r"a" 'e "r/a = T .r'" Y f a-^^-^e-^'+'da^lx, Jo (l+x)"^" 114 INTEGKAL CALCULUS. [AUT. 99. Therefore ^i^il') = f ^"" dx ; [4] r(m + H) Jo (1 +0;)"'+" ■- -^ or by [3], ^ ^ Jo ^ ^ r(m4-w) If H = l— m, tlieu since r(l)=l, f '_J— _ dx = r" ^^ (/.r =T(m)T(l~ m) . [61 Jo {i-xy Jo 1+a; y ^ y ) l j Formula [G] leads to an interesting coulirmation of Art. 92(5). Let m — ^, and we have from [(>] Substitute 2^ = V-'"' and we have I ^ = 2 I — -— = ir. Jo (l+a-)V-f Jo 1+2/' Hence ra)=V'^; [7] and since by Art. 97 (c) jf%'V/.x- =^r(^), 99. By the aid of formulas [4], [a], and [7] of Art. 98 a number of important integrals can be obtained. For example, let us consider I " sin".i'.'/.r, where n is gjicater than — 1. Let // = siiix-, and bin" x.dx = I ?/"(!— y-y'Hly. «/o Chap. VIII.] DEFINITE INTEUUALS. 115 Let, now, ^ = ff-, and But b(^^\ 1^ = V I ) by [.>] Art. 98. by [7] Art. 98. ^i$ -) Hence C\ur. ,,. = -/'- ±J-4. [I] Jo 2 j^fn , i\ rf^ + 1 If n is a whole number, this will reduce to the result given in Art. 87, Ex. 5. EXAMPLKS. (2) I sin".'ccos'"a;.c/u; = — ^^ — - — r — 2rf:^ + i IIG INTEGRAL CALCULUS. [Akt. 99. (3) /'-^ _v^ i) Vi — x" " ^C>l) r(i' + i)r r ;. + ! + (4) rV(l-a;'')''(/.t=- - >/i + m + 1 CiiAi'. IX. J LENGTHS Oi'' CUKVES. 117 CHAPTER IX. LENGTHS OF CURVES. 100. If we use lecttinguhir eoordiuates, we have seen (I. Art. 27) that tanr = g, [1] and (I. Arts. 52 and 181) that (Is- = (J,i- + dy\ [2] From these we get siur = -^i [3] [4] COSr = — , (Is by the aid of a little elementarv Trigonometry. These formulas are of great importance in dealing with all properties of curves that concern in an}- way tlie lengths of arcs. We have already considered the use of [2] in tlie first volume of the Calculus, and we have worked several examples by its aid in rectification of curves. Before going on to more of the same sort we shall find it worth while to ol)tain the equations of two very interesting transcendental curves, the catenari/ and the tractrix. The Catpnary. 101. The common aUpiutn/ is the curve in which a uniform heavy flexible string hangs when its ends are support<'d. As tlie string is flexible, the only force exerted by one portion of the string on an adjacent portion is a pull along the string, which we shall call the tension of tlie string, and shall n-present by T. T of course has different values at dilfcrcnt points of the string, and is some function of the coordinates of tlu' i)oiiit in question. 118 INTEGRAL V)ALCULUS. [Art. 101. The tension at any point has to support the weight of the por- tion of the string Ik'Iow tlie point, and a certain amount of side pull, due to the fact that the string would hang verticalh' were it not that its ends are forciljly held apart. Let the origin be taken at the lowest point of the curve, and suppose the string fastened at that point. Let s be the arc OP, P being any point of the string. As the string is uni- form, the weight of OP is proportional to its length ; we shall call this weight ms. This weight acts verti- cally downward, and must be balanced by the vertical effect of T, which, by I. Art. 112, is T^sinr. Hence Ts\nT = ms. (1) As there is no external horizontal force acting, the horizontal effect of the tension at one end of any portion of the string must be the same as the horizontal effect at the other end. In other words, TcosT = c (2) where c is a constant. Dividing (1) by (2) we get = — tan T, 'ni where a is some constant, tion in terms of x and y. From this we want to get an equa- hence or and tanr = Vsec^T — 1 = \ -rr, -«'(r^•)• ads (a' + s')^ = dx. Integrate both members. CiiAi>. IX.] LENGTHS OF CURVES. HO a \og{s + VoM^ .s~) = ./• -I- C ; when a;= 0, s= 0, hence C=aloo[a, and log (s 4- Vo- + .s") = '- + log a, s + -\/a--{- tr =rte«, Va^+ «" = <^fca — s, s = - (e« — e «) = a tan r by (3) . ; I Hence a^= - (e^- e"!), i and ?/ = ^ (('a -f- (?-!) + C. 1 If we change our axes, takhig the origin :it a point n units i below the lowest point of the curve, y = a when x = 0, and i therefore (7=0, and we get, as the equation of the catenary, '• Example. ; Find the curve in which tiie cables of a suspension-bridge ; must hang. 7>*-^7';„*^r Aiis. A i)aralu)la. A^T^ T/»- r,™/,«. ^^-, „^^.^ 102. If two particles are attached to a striilg, and rest on a-^* * rough horizontal plane, and one. starting with the string stretclied, moves in a straigiit line at right angles witii tlie initial iiosition of the string, dragging tlie other particle after it, the patli of the , second particle is called the trartrix. Take as the axis of X the path of the fii-st particle, and as ^ the axis of Y the initial position of the string, and let a be ; 120 INTEGRAL CALCULUS. [Akt. 102. the length of the strinof. From the nature of the curve the string is always a tangent, and we shall have for any point P y = — SUIT, (( for r lying in the fourth (juadrant has a negative sine. Y [1] hence and cr asr dx^ + dy^ y\d.r + dy-) = ahly\ yhlx- = {ii.(hi — 0, , x.dx =«I dx V«- — af dx f" dx ira The length of a (juadrant = a I = — ; ,-. the length of a circumference = 2 7ra. 122 INTEGRAL CALCULUS. [Art. 104. Length of Arc of Cycloid. 104. For Hie c^'cloid we have X = aO — a sin ] ^y- (I. Art. 99.) ?/ = « — acos^J dx = (i{l — cosO)dO = yds, $ = vers -i.V de=^ ''.'/ dy " ^i-yy y' V2«y/-/ \ a a^ dx = ?^dy V2 ay — y- CZ6-' = dx^ + df = ^^^^^ = '^«^y\ 2 ay — ?/- 2a — ?/ ds = V2 a dy V2 a — y s = V27( r'^=^= = 2V27t(V2;rr7o - V2;r:^) »/»o V2« — ?/ If the arc is measured from the cusp, ?/„ = 0, s = 4 a - 2 V27i V2 a - y,. [1] If the arc is measured to the highest i)oiiit, ?/i = 2 a, .s = 4a. The whole arch = 8«. Example. Taking the origin at the vertex, and taking the direction down- ward as the positive direction for y, the equations become X = ((6 + a sin 6 (I. Art. 100.) y = a — ac ■ ■ Show that s = 2yj2tiy when the arc is measured from the summit of the curve. Chap. IX.] LENGTHS OF CritVKS. 123 10."). "We can rectify the cycloid without climiiiatiiij; B. X = ad — a sin ( y = a — ocosi dx^ + d;/- = 2 a-de'i 1 - t'os^) , and s = a^2 i (1 -cos$)\d6. If ^y = and 6i = 2 tt, we get .s = 8a as the whole curve cos - 2 106. Let ns find the loif/lh of an arch of the ejifci/cloid. X = {a + h) cos^ - b eos^-^^-±-^^ I (I.Art.lo:»[l].) b dx = dy = ds' = (a+by'dff („ + /,) shi 6 + {a + b) sin'-^^l {a + b) cos^ - {a + b) cos^^-±-^'^l dd. dO. r|2- 2 (c-os'^'^ cos^ -\- -sin'-^'^ sin^^l s = (a-\-b)^/2 C^'fl-vosjOy'ie, 4b(a +b)r a ^ ^i n~\ rn 8= ^ ^ q eos — ^o-cos — 6, ■ [1] a |_ 2^> 2o I To get a coinpU'te arcli we must h't ^„=ii and ^, = :i- n-. «/ Hence, for a whole arch, Hb{a +b) 124 INTEGRAL CALCULUS. [Art. 107. Examples. (1) Find the length of an arch of a hvpocycloid. ' / 8b(a-b) Ans. s = . a (2) Find the length of an arch of tiie curve x^-\-!/^ = a^^ and show that it agrees with tie result of Ex. 1. (v. I. Art. 109, Ex. 2.) 107. Let us attempt to find the length of an arc of the behave ^ x ./.. _^ 2 ;/ r^^ ^ a- h- b-x , cry a*y^ u^ — XT a2-eV a^-a^ dx". where e is the eccentricity of the ellipse. The length of the elliptic quadrant is These integrals cannot he olttained directly, but [a^ — eV\^ can be expanded by the liinoniial Theorem, and the fractions formed by dividing the terms of the result by \_a^ — x-'\^ can be integrated separately, and we shall have the required length expressed by a series. A more convenient way of dealing with the problem is to use an auxiliary angle. Instead of \, 4" '.^ = 1 we can use the pair of equations X = a sin .-r . . ~^ , ^ (I. Art. 150), y = 6C0S<^J ^ ^' Chap. IX. ] LENGTHS OK CURVES. 12.') dx = a cos(f>.d(f}, dy = — b sin (f>.d(f>, ds^ = (a'-cos-<^ -f- bH\n-)d- = [a^ - (a- - b^^Hm-tfyldffr = a-fl — ^^~^" sin-<^') f?<^- = a-(l - e-.siir-»y-d. [4] Example. (1) Obtain s, as a series from [2], and also fVoin [4], and compare the results with Art. 1>1, Ex. ."). Polar Formulce. 108. If we use polar coordinates we iiave ds = \l~d,^ + r- d4-, ( I . A rt. -20 7 , Ex . 2 . ) tanc = ^, (I. Art. 207.) dr From these we get, l)y Trigonometry, sine=^-^, cos€ = — . ds (i-'i 109. Let us find the equation of the curve which crosses all its radii vectores at the same angle. Here rdd) adr ., taue = a, a constant, --^ = a, —- = a' rd4> = adr, ' ds^^dr + ?"(?(^2 = (1 + a^^di^ ; s = C{\ + (r') i (?/• = ( 1 + c(-) h(r, - r,) . Examples. (1) Find the length of an arc of the parabola from its polar equation r= 1 -f- cos 4> (2) Find the length of an arc of the Spiral of Archimedes r = Uff). 111. To rectify the Ca/rZ/oiYZe. We have r=2a(l-cos<^), (I. Art. 109, Ex. 1), dr= '2as'u\(f).d.d(j)- + I «-{] — coH)-dcf>^ = 8 ct-d(f)-{ 1 — cos <^) , s = 2 ^2 . a ^i\ •-cos<^)V/<^ = 8 afcos^ -cos|n = lOff for the whole perimeter. Chap. IX.] LENGTHS OK CriiVES. Involutes. 112. If we can expi-ess the leugth of the arc of a <;ivon curve, measured from a fixed point, in terms of the eourdimites of it.s variable extremity, we can lind the equation of the involute of the curve. We have found the equations of the evohite of y=fj: in the form x' = x — p cos V y' = y-p sin V (I. Art. 1)1). We have proved that tani' = tanr', (I. Art. 'J.j), fit' and that 3^=1» (I. Art. UG) ; dp smr = -^, els' , dx' cos T = -— ds' . (Art. 100). Since tanv = tanT', v = t' or r=l.sO° + T'. As normal and radius of curvature have opposite directions, we shall consider v = 180° + r'. Then sin i' = — sin r' and cos r = — cos t'. Hence x' = x-\- p-j-i 0) ds Since dp = ds\ P = s' + l (3) where /is an arliitr.irv ((.nstmit. Since ./• and // are the coordinators of any point of the mvohtte, it is only necessary to eliminat*.' x', y', and p by combining equations (1) . (2), and (."5) with the equa- tion of the e volute. As we are supposed to start with the e(iuation <>f the evolute and work towards the equation of the involute, it will be more natural to accent the letters belonging to the latter curve instead 128 INTEGRAL CALCULUS. [Art. 112. of those going with tlic former ; and our eqnations may be written X = x'+p''^; y = y'+p''k; p'=S + L (4) as as Since p'=l when .s = 0, it follows that I is the free portion of the string with which we start. (I. Art. 97.) By varying I we may get different involutes of the same curve. To test our method, let us find the involute of the curve for which / = m. We must first find s. 2ychj = -^{x-m)-dx, 9 m y , ., 2 X -\- ill , o (Is- = — dx-, "dm ■ , 1 V V3 m^m ;} V 3 m <-^ p = .s- -\- m = - ; V3 m -'-n m 2 / m {•2x + m){x-m.Y y , X — ni. X- .J . .T = 3.r'-f-m, V Substituting in (.^>) tl le values of x and y just obtained, we have y^ = 2 ?».»•' the equations of the rcfiuwed involute. Ciivi' IX.] LENCTHS OF CURVES. 1 'il* Example. Find an involute of aif = x^. U.S. An involute of the cycloid is easily found. Take ecjua- lionsl. Art. 100(C). X = aO + a sin | y = —a-\-a cos 6 ) Let p' = s, dx = a ( 1 + cos 6)de =2 a cos- ^ cW, 6 9 dii = — a sinO.iW =— 2 a sin - cos - (W, J 2 2 els' = 2 o2 dd- ( 1 + cos 6) = A a- dO- QoA s = 2aj cos-(/^ =4asni^^' 6 6 X = x' + 4 a sin -cos - = x' -\--2a sin 9, Q y = y' — ia sin-- = y' — 2 (< ( 1 — cos ^) , x' = a^ — (I sin < y' = a — (( cos < a cycloid with its cusp at the summit of the given cycloid. Example. "^ From the equations of a circle x = a cos < y = a sin < obtain the equations of the involute of the circle. Let / = (» Ans. x'= a(cos0 + sin <^) y'= a (sin «/> — ) 130 INTEGRAL CALCULUS. [AuT. 114. Intrinsic Equation of a Curve. 114. An equation connecting tlie length of the arc, measured from a fixed point of any curve to a variable point, with the angle between the tangent at the fixed point and the tangent at the variable point, is the inti'insic eqiiation of the curve. If the fixed point is the origin and the fixed tangent the axis of X, the variables in the intrinsic equation are s and t. We have already such an equation for the catenary s=atanT, Art. 101 (3), [1] the origin being the lowest point of tlie curve. The intrinsic equation of a circle is obviously s = ar, [2] whatever origin we may take. The intrinsic equation of the tractrix is easil}' obtained. Vie have 2/ = -asinT, Art. 102 (1), and s = a log- ; Art. 102, Ex. 1. y hence s = alog(— csct) where t is measured from the axis of X, and .9 is measured from the point where the curve crosses the axis of Y. As the curve is tangent to the axis of Y, we must replace t by t — 90°, and we get s = alogsccT [3] as the intrinsic equation of the tractrix. Example. Show that the intrinsic equation of an inverted cycloid, when the vertex is origin, is s = 4asinT; (1) when the cusp is origin, is .s = 4o(l — cost). (2) CiiAi'. IX.] LENGTHS OF CI HVKS. l;jl 115. To find the intrinsic equation of the epicveloiil we can use the results obtained in Art. lOG. di,==(a-\-b)feosO-i'os^-^^e\w=2(a+b)sm'^^^$Hm — 6.(10, \ ^> J '2 b 2 b by the formulas of Trigonometry sin a — sin^ = 2 cos ^ (a + /?) sin^(a — ^), cos/S— cos a = •2sin^(u + (3) sin ^(u — /3) ; tan T = — = tan — ■ 6, dx 2 b ' , a + 2b^ hence t = — — — ; 2o ^ ^ ^ ' 1 — cos i^^)byArt.ir)r,[i; therefore s = ^-^^^^^±^H \ - cos -^ r) [1] a \ a + 2b J is the intrinsic equation of the epicycloid, with the cusp as origin. If we take the origin at a vertex instead of at a cusp ^_^(a + 2b) 2 a + t' ' „, 4h(a+?>) sin- a a a + 2 b „_ib(n+b) sin- a — T or s = — - — ^i— ^sin — -T I 21 (I (( + 2b is the intrinsic e(iuati«)n of an epicycloid ri-ferred t^ "^yr" ^ ds ds ds ^ ^r ^ hence ds = sec tF't.^/t. a/ S ^ f-t^-^"^ ^-^^A'-) Integrating both nienilx'is we shall have the reqiured intrinsic equation. For example, let us take x^ = 2 my, which is tangent to the axis of X at the origin. '^^ •Ixdx^'lmd]!, '*'' a^ « ^J ((X= VlSQC''T.ClT, ^ . a>-^^^^ = tan 7 = X m r^.r = VI SCi/r.dT, dx ds = COSt = m sec^ r — (7.S ds = ?» secV.f?r, *-- n^ CX^ ds =msQi-'^T.dr, (1) = 0; .-. C'=0; g^^^^^^-^-f^^ pf^ ^ Examples. (1) Devise a method when the curve is tangent to the axis of I", and apply it to ?/*= 2 ma;. (2) Ohtain the intrinsic equation of v'-' = (x — wY- 'It in (3) Obtain the intrinsic equation of the involute of a circle. (Art. 113, Ex.) 134 INTEGRAL C^VLCULUS. [Art. 118. 118. The evolute or the involute of a curve is easily fouucl from its inti-insic equation. \ si i r'N \(2) p. J--^/t If the curvature of the given curve decreases as we pass along tlie curve, p increases, and s' = p-p^. (I. Art. 9G). If the curvature increases, p decreases, and Hence ahva^'i ,s =p„ — p. s' = ±{p — po); [1] p = — , (I. Arts. 86 and 90). (It We see from the fi a ih ■ih ((I -\-b) .• a r,T s = ^^ — — -sni r. ri| a + 2 1> (i + -2b "- -■ The form of [1] is that of an epicycloid referred to a vertex as origin ; let us lind ', the radii of the lixed and rolling circles. 4b'(a' + b') . a' , » , , , - r.>i . s = ^^ ■ ^sni -T, l)v Alt. 1 1.» [2] ; , , hence, 4 6V + //) ^ 4/. (. + /.)^ ^-^Jj^^jLllI dC-h •v; r/' a i^^H^ ^ K o 136 INTEGRAL CALCULUS. [AuT. )2L Solving these equations, we get a + 2b' h' ab a + 26 a' h' a ~b' and the radii of the fixed and rolling circles have the same ratio in the evolute as in the original epic3-cloid ; therefore the two curves are similar. Example. Show that the evolute of a Inpoc^'cloid is a similar hypo- cycloid. 121. We have seen that in involute and evolute r has the same value ; that is, t — t'. If s' and t' refer to the evolute, and s and t to the involute^ we have found that 8' = —!'', or s' = — ^ — /, / being a constant, cIt the length of the radius of curvature at the origin. {s' + l)(W = ds, is the equation of the involute. The involute of the catenary s = a taur is, when / = 0, <( I tun T.(ir = a log sec T, the tractrix. Chap. IX.] LENGTHS OF CUllVES. 137 The involute of the cycloid .s = 1 d siiir wlieii / = i.s s = 4a i smT.dr = 4(i{\ — cost), an equal C3'eloid referred to its cusp as origin. The involute of a cycloid referred to its cusp .•<= 4*^(1 —cost) when / = is .s = 4a I (1 — cosT)f?r = -la(T — siuT), a curve we have not studied. The involute of a circle .s = ar when I = is =«i 122. While any given curve has but one evolutc. it has an infinite number of involutes, since the equation of tlic involute -^■=X\s + nh = !- + — + f f'./V.fM i/ii .'0 " 2 »'" .'" is the second involute ; . = /T + ^ + ^^ + +'i:+f\r,lr-^ (2) 2 .3 ! u I . '0 is the nth involute. 138 INTEGRAL CALCULUS. [Art. 123. By Maclamiii's Theorem, But s = when t = ; hence fo = 0, and ./V = .l,r +i,= + £!|..+ o ! 4 ! as n increases indefinite!}' all tlie terms of (3) approacli zero (I. Art. 133) , and tlic limiting form of (2) is s = ir + ^r + ^ + 2 ! 3 ! = ?(^1+- + — + — + -1 V 1 2! 3! s = l{e'-l) by I. Art. 133 [2], which is a logarithmic spiral. 123. The equation of a curve in rectangular coordinates ib readil}- obtained from the intrinsic equation. Given «=./V, we know that sin t = — , (Is and cos T = — ; (Is heuce dx = cos rds = cos t/V.^t, dy = sin rds = sin rfW.dT^ a; = I cos t/'tAt \ y:=p,Urrr..lr\ Chap. IX ] LENGTHS OK CURVES. 139 The elimination of t I)i'twecn these equations will give us tiic equation of the curve in terms of x ami y. Let us apply this method to the catenary. s = « tanr, ds = a sec^T.dr, X = a fseer.f?r = a log x'^^ti^ULT, Jo \ 1 — sinr y = (I i sec T tan r.dr = a (sec t — H , 1 + sinr 1 — sinr' 2z X e " — 1 e" — e' e « + 1 f^« -f- e a the equation of the catenar}- referred to its lowest point as origin. Curves in Space. 124. The length of the arc of a curve of douhle eiu-vature is the limit of the sum of the chords of smaller ares into which the given arc maybe broken up, as the number of these smaller ares is indefinitely increased. Let (x,y, z) , (x + d.v, >/ -}- A//. 2 -}- Az) be the coordinates of the extremities of any one of the sma ll ares in question; c?a;,A//, Az are infinitesimal; Vr/af^-|-A//-'-h Az^ is the length of the chord of the arc. In dealing with the limit of the sum of these chords, any one may be replaced by a (piantity dif- fering from it by infinitesimals of higher order than the first. y/dx- + dy-+ dz' is such a value ; hence 140 INTEGRAL CALCULUS. [Art. 124. Let us rectify the helix. X = a cos 6 1 ^J = asinO L (1. Art. 214.) z = kd J dx = — asinO.dO, dy = a cos6.dd, dz = kde, ds^ = {a^ + Jr)d6\ s = {a- + A-) i ( "\W = \/a^+k\Oi - 0,) . Examples. (1) Find the length of the curve (y = — , z = ^\ V -la Gay Ans. s = x-j-z-\-l. (2) y = 2 \/ax - a;, 2 = a; - | J-. Ans. s = x + y-z-\-l. CriAi-. X.] 141 CHAPTER X. 125. We have found and used a formula for the area bounded by a given curve, the axis of X, and a pair of ordiuates. =fydx. We can readily get this formula as a definite integral area in the figure is the sum of the slices into which it is divided by the ordinates ; if ^x, the base of each slice, is indefinitely decreased, the slice is infinitesimal. The area of any slice differs from y\x by less -than A?/Ax, which is of the second order if Ax is the principal infini- tesimal. We have then r. Mie . limit ^,"^1 ^ ^ = ^x=o ^ y^"" Hence -=X' ydx. bv I. Art. ir.l, [1] If the curve in question lies above the axis of X, and r^ is less than a^i, each ordinate is positive, each Ax is positive, each term of the sum whose limit is required is positive, the .sum is positive, and the limit of the sum or the area sought is positive. If, however, the curve lies below the axis of A', and .r„ is less than x,, each ordinate is negative, each Ax is positive, each term of the sum is negative, the sum is negative, and the limit 142 LNTEGiiAL CALCULUS. [Alir. 125. of the sum or the area sought is negative If, then, the curve happens to cross tiie axis between .i-y and x'l, Ibruuila [Ij gives us the difference between the portion of the area above the axis of X and the portion below the axis of X, but throws no light upon the magnitudes of the separate portions. Consequently, in any actual geometrical problem it is usually necessary to find the portion of the required area above the axis of X and the portion below the axis of X separately ; and for this purpose it is essential to know at what points the curve crosses the axis. Indeed, if the problem is in the least complicated, it is neces- sary to begin by carefully tracing the given curve from its equation, and then to keep its form and position in miud during the whole process of solution. Examples. (1) Show that I \'ch/ is the area bounded by a curve, the axis of Y, and perpendiculars let fall from the ends of the bounding arc upon the axis of Y. (2) If the axes are inclined at the angle w, show that these formulas become \ A = sin w I ydx = sin w | xdy. (3) Find the area bounded by the axis of X, the curve ar^ + 4_v = 0, and the ordinate of the point corresponding to the abscissa 4. Ans. b\. (4) Find the area bounded by the axis of X, the curve 2/ = ar', and the ordinates corresponding to the abscissae —2 and 2. Ans. 8. (5) Find the area bounded by the axis of X, the axis of Y, the curve .v = cosa;, and the ordinate corresponding to the abscissa Stt. Ans. 6. Chap. X.] AUKAS. U: 120. lu polar foordi nates wc cixn regard the area 1)etween two radii vectores and the curve as the Hinit of the sum of sectors. The area in question is tlie sum of the smaHer sectorial areas, any one of which differs from ^ >-^A<^ In- less than tlie ditlerence between the two circular sectors ^(r + A;-)'''A<^ and ^rA<^; that is, by less than (A/-)^'A(/> 2 /•A;-A. 127. Let us find the area between the catenary, the axis of X, the axis of Y, and any ordinate. A = Cydx =- i(e^ -\-e'a)(lx, but Hence A = ^{ea-e-a), (C« — €") = by Art. 101. A = a.5, and the area in question is the length of the arc multiplied by tlu- distance of the lowest point of the curve from the origin. 128. Let us find the area l)etween the ti-actrix and the axis ofX. We have (^!/. A = Cydv = - \(hj\l(r - if. (.\rt. 102.) 144 INTEGHAL CALCULUS. [Art. 129. The area in question is .'.. 4 iiicli is the area of tlie (juadrant of a circle with a as radius. Example. Give, by the aid of infinitesimals, a geometric proof of the result just obtained for the tractrix. 121). In tlie last section we found the area between a curve and its asymptote, and obtained a finite result. Of course this means that, as our second l)ounding ordinate recedes from the origin, the area in question, instead of increasing indefinftelj, approaches a finite limit, which is the area obtained. "Whether the area between a curve and its asymptote is finite or infinite will depend upon the nature of the curve. Let us find the area between an liyi)erbola and its asymptote. The equation of the h} [)erbola referred to its asymptotes as axes is -' 4_ / 2 Let (D be tiie angle l)etween the asymptotes ; then A = sin (J) I yax = sm (» I — = ao .yii ■ 4 J(i X Take the curve y'x = -i i 'jjdx Ol- .1 = 2 i 'j;hi ; by tlic first formula. by tlie second. ■'-■ (hi Jo y- -\- A a- EXAMPI.KS. (1) Find the area between the curve y-{x- + a-) = (rx^ and its asymptote y = n. Ans. A = -la'-. (2) Find the area between y"('2a— .r) = .r'' and its asymptote x = 2 a. Aus. A = :) -((-'. f3) Fincl tlie area bounded by the curve y-=— — ' and . ~ , , (I — X Its asymptote x = a. Ans. A = '2crf\ + -\ 130. If the coordinates of the points of a cnrxc iin- ex- pressed in terms of an auxiliary variable, no new dillit iilty is presented. Take the case of the circle x'- -\- y- = a'\ whicli may be writttMi .r = ((c()s0 i y = a sin cf> ) dy = (icofi(J(fi. The whole area A = a- I cos-(f)d4> = tto-. 146 INTEGRAL CALCULUS. [Art. 13L Examples. (1) The whole area of an ellipse ^ — "'^^^'P l jg ^^5, y = sin ) (2) The area of an arch of the cycloid is d-n-a^. (3) The area of an arch of the companion to the cycloid = a$, y = a{\ — cos^) is 2 ttcl^. 131. If we wish to find the area between two curves, or the area bounded by a closed curve, the altitude of our elementar}- rectangle is the difference between the two values of y, which correspond to a single value of x. If the area between two curves is required, we must find the abscissas of their points of intersection, and they will be our limits of integration ; if the whole area bounded by a closed curve is required, we must find the values of x belonging to the points of contact of tangents parallel to the axis of Y. Let us find the whole area of the curve ci^y- + h'-x^ = a-lroi?, or a*y- = lrx-{a^— x^) . The curve is sj-mmetrieal with reference to the axis of X, and passes through the origin. It consists of two loops Avhose areas must be found separately. Let us find where the tangents are parallel to the axis of Y. b (ly _ h ((- — 2 x^ (Ix 'a- — X- = tanr. : - when tanr = », that is, when x — ± a. 2 yi = 2 - fx V((^ - x'.dx + 2 -, Cx Va- - x'.dx = | ab. Cn\T. X.] AREAS. 147 Again ; find the -whole area of (^y —xy = a^ — 7?. y = x± ^Id- — ar, A = C{y'- y") dx = p2 V^?^^' . dx. nits of integratic ^ _ Va- — x^ q: x To find the limits of integration, we must see wiiere t = -• 2 dx y/ar — XT = X when x = ± a. ^ = 2 I V((" — .r . (?a* = tto- KXAMPLES. 3r ^fl -4' X^ (1) Find the area of the loop of the curve ?/-= ^ a — X Ans. 2 a- ( 1 — 4 (2) Find the area between the curves y- — 4ax=0 and ^ ^l»s. 3 (3) Find the whole area of the curve x^ + v^ = a}. Ans. fjra^ 4 a- (4) Find the area of a loop of d-y^ = x\a- — a-) . ^-l«.s-. -^— o (5) Find the whole area of the curve 2 / (a^ + ar') - 4 ay (a^ - ar) + ((«- - ar)= = 0. Ans. (rVA-^Y 132. We have seen that in polar coordinates A = ^\ rd. Let us try one or two examples. (a) To find the whole area of a circle. The polar ecpuition is /• = a. A = \j'''' = 7raK 148 INTEGRAL CALCULUS. [Art. 132. (/') To find Ihe aiva of the cardioide r= 2ff(l — cos<^). ^1 = ^ I 4 a- ( 1 — cos Yd(i> = 2 o- 1 ( 1 - 2 cos <^ + cos' (l(f>)d = ■-' ^ X 2 , J . xfhi— ydx sec'cf>dcf>= ■' y ; 1)nt, since x = r cos <^, sec = -; X hence i^^^xdy-ydx^ and r- d(j> = xdif — ydx ; dx = (a + b) f- sin ^ +sin ^-^-±-^ ^^ f/^, r/v = (a -\-b)f i:ose - cos'-L±J!e]de. xdy - ydx = (a + b) {a + 2 b)f\ - cos - Aw = i'^dcfi. Our limits of integration are olniouslv and — ^. niAi'. X.] AREAS. 149 Hence A = ^{((-i-b){((+-2 h )J ' " / 1 - fos - 6\ de. A = ^{a+b){u+2b), is the area of the sector of the epieycloiil. Siihtnut the :uv;i of the circular sector Trab, and we get -1 = IT a as the area in question. (d) To find the area of a loop of the curve r = a-cos 2 . For any value of cf) the values of /• are equal with opposite signs. Hence the origin is a centre. When<^ = 0, r=±(i; as increases, r decreases in length till <^ = -, when r = ; as soon as 0> -, r is iniaginarv. If d> 4 4 decreases from 0, /• decreases in length until 0= — -. when r = ; and when <^ , r is imaginary. To get the area of a looj), then, we nuist integrate from (f)= —^ to cf> = -. A = ^jr^^H = ^ crf'^vo^ 2 ct>.dct> = ^'. EXAMPLKS. (1) Find the area of a sector of the parabola r 1 4- cos (2) Find the area of a loop of the curve rcoscf> = irs'm :5 0. Itis. '-^ '" log 2. I (3) Find the whole area of the curve r = (i(vos'2 + tan<^) ami its asynip- tote rcos<^ = 2((. .,,,^ (--hAc'- \'^ J 150 INTEGRAL CALCULTS. [Akt. 138. 133. Wliiu the eqiuiUou of a curve is given in rectangular coordinates, we can often simplify the problem of finding its area by transforming to polar coordinates. For example, let us find the area of {x' + y-)- = ia-x^-\-4bh/. Transform to polar coordinates. '/•■' = 4 r(a2cos2<^ + b- H\n^), t'^ = i {(i^ cos- = 2 7r(a- + b-). Examples. (1) Find the area of a loop of the curve {x- + y-y = 4 a-x^ij^. Arts. — — (2), Find the whole area of the curv x^ ?/2 1 /.r2 w^' a*^b* c\a'^by Ans. ^(a' + b'-). 2ab (3) P'ind the area of a loop of the curve y^ — S axy -f x^ = 0. 2 ' A71S, 134. Tlie area between a curve and its evolute can easily be found from the intrinsic equation of the curve. It is easily seen that the area hounded by the radii of curvature at two points infinitely near, by the curve and by the evolute, dif- fers from ^p^f/r by an infinitesimal of liigher order. The area bounded l)y two given radii vectorcs, the curve and the evolute, is then CllAP. X.] 151 Hence uT'^)- For example, the area between a cycloid anil its evoliite is Let = 8a^ I cos-TfZr. and T. = =«"! 2cos-tcZt = 'Zira^. Examples. (1) Find the area between a circle and its evohite. (2) Find the area between the circle and its involute. r^ Ilolditch's TJieorem. 135. If a line of fixed length move with its ends on any closed curve which is always concave toward it, the area between the ^ curve and the locus of a given point of the moving line is equal to the area of an el- lipse, of which the segments into which the line is divided by the given point are the semi-axes. Let the figure represent the given curve, the locus of /^ and the envelope of tiie moving line. Let .l/*=a and Pn = f>, and let Cli = p, C Iteing the point of contact of the moving line with itis envelope. I>ct AB = a + b = c. 152 INTEGRAL CALCULUS. [Art. 135. The area between the first euvAe nnrl the .second is the area between the fir.st curve and the envelope, minus the area between the second curve and the envelope. Let 6 be the angle wliich the moving line makes at any instant with some fixed direction. Let the figure represent two. near positions of the moving line ; A^, the angle ])etween these posi- tions, being the principal in- finitesimal. PB = p, P'B' = p + Ap. The area PBB'P'P diflfers from ^p~d6 bv an infinitesi- mal of higher order than tlie first. ^p-cW is the area of PBMP, and differs from PP'NB by less than the rectangle on PJ/and PQ, which is of higher order than the first, by I. Art. L53. But PP'NB differs from PP'B'B by less than the rectangle on BX and NB\ which is of higher order than the first, since NB\ which is less than I*P'-\-\p, is infini- tesimal and A6 is infinitesimal. The area between the first curve and the envelope is then hW ; or, since we can take PP'A'A just as well for our elementary area, ^ I {c — p)^cW. U}'dO=^r{c-prdO; Hence whence 2 c rpf/^ = 2: ^2rr I pdd = ir<. (1) The area between the second curve and the envelope is ,2Tr hj^ip-hyae. Chap. X.] AREAS. 1, The area between the first curve and the .second is then l.y (1), = bjjj(W-b'7r = Trbc — b'- TT = 7r6(o -hb) — b-ir, A = 77Ub. (2) which is the area of an elUpse uf wiiich a and b are serai-axes. Q. E. D. Examples. (1) If a line of fixed length move with its extremities on two lines at right angles with each other, the area of the locus of a given point of the line is that of an ellii)se on the segments o( the line as semi-axes. (2) The result of (1) holds even when the fixed lines are not perpendicular. Areas by Double Integration 136. If we take x and ?/ as the coihdinates of any point P within our area, x and y will be independent variables, and we can find the area bounded by two given cui-ves, // = ./> and // = F.r. by a double integration. Supposo the area in question divided into slices by lines drawn parallel to the Mxis of y, anil these slices sulttli- \ ided into parallelograms by lines drawn parallel to the axis of X. The area of any one of tiie small parallelograms is A'/A.r. If we keep X constant, ami take thr sum of these rectangles from y=fx to »/ = ^'f- we sliali get u result differing from the area of the corresponding slice by less than 154 TNTEGRAL CALCULUS. [Art. 137. 2Aa;A?/, which is infinitesimal of the second order if Ax and Ay are of the first order. Hence ^x.dy = Ax t dy is the area of the sUce in question. If now we take the limit of the sum of all these slices, choosing our initial and final values of X, so that we shall include the whole area, we shall get the area required. Hence A= i '( i dy\dx. In writing a double integi'al, the parentheses are usually omit- ted for the sake of conciseness, and this formula is given as (h\ and A. st sum our rectangles, le area of pf<''f' i d. and A= i ) nlifxlr. Or we may first sum our rectangles, kee[>ing /• luiehauLred. and we get as the area of efi'\f (2) 156 INTEGRAL CALCULUS. [Art. 138. It must be kept in mind that r in (1) and (2) is the radius vector of any point within the area sought, and not of a point on the boundary. For example, the area between two concentric circles, r = a and r = h, is .1 = r Crdcfydr = C Cnlrd^ = ir^a^ - 6^) . Again, let us find the area between two tangent circles and a diameter through the point of contact. Let a and h be the two radii, ?'=2acos<^ (1) and r=26cos^ (2) are the equations of the two circles. J2 h cos .70 2 If we wish to reverse the order of our integrations we must break our area into two parts by an arc described from the origin as a centre, and with 2 6 as a radius ; then we have = f Crddr+ C Crdxfxir Jo J r Jib .'0 cos-' 26 r( COS"'— - — cOs~'— - )f7r-|- j?-cos~^T-dr V ^a 2b) J2b 2a |(a^-6^). EXAMPLK. Find the area between the axis of X and two coils of the spiral /•= aN^ the same arc when it revolves A'' about O'X'. Let «S' be the surface about OX, and 6" about O'X'. ■ ^Ve have S = 27r f'yds, »S"= 2 tt C'i'ds'. 160 INTEGRAL CALCULUS. [Akt. 141 By Anal. Geom., x = x\ y = !h + y'- Hence (Jx = dx\ chj = df/', ds = ds\ and S=2-!r i {ijo + y') d-s = 2 Tri/o{-^i — .s,,) + iw | y^ds, = 2 7r?/o(.s', -So)4-'S". There fore S ' = ,S' - 2 tt.Vo ( Si - -So) . [ 1 ] Si — .S) is the lengtli of the revolving curve ; 2 tt?/,, is the cir- cumference of a (;ir(;le of which y^ is the radius. Hence the new area is equal to the old area minus the area of a cylinder whose length is the length of the given arc and whose base is a circle of which the distance between the two lines is radius. In using this principle careful attention must be paid to the sign of yo, and it must be noted that the original formula s will always give a negative ^'alue for the area of = 2 TT ( yd Js,, the surface generated, if the revolving arc starts from below the axis ; and hence, that the surface generated by the revolution of any curve about an axis of symmetry will come out zero. As an example of the use of the princi- ple, let us find the surface of a ring. Let a be the distance of the centre of the circle from the axis, and h the radius of the circle. vSince the area generated b}- the revolution of the circle about a diameter is zero, the recjuired area is 'lirhrlira — \irah. EXAMI'LK. Find the area of the ring gciu'rated by the revolution of a cycloid about any axis parallel to its l)ase. Ans. o = 4 uhTT[ it -\ oh CiiAi'. XI.] AIIKAS OF srUKAC^ES. 142. If we use polar for»rcliiKxtes, r sin • the revolution of the upper half of a eardioide about the hori- zontal axis. r = 2a(l — cos(^) ; dr ='2 a sin<^.rf<^, ds2=8a-(l-eos<^)f?<^2^ S = -2tv Ca-^2 a-{ 1 - cos <^) ^ sin .d(f>. P^XAMPLES. (1) Find the surface of a sphere from the polar equation. (2) Find the surface of a paraboloid of revolution from the polar equation of the parabola m 1 — cos C;/lin drica I S u rfaces. 143. If a cylindrical surface is generated bv a line which is always parallel to the axis of Z, the area of the portion bounded by two positions of the generating line, the plane of A'l', and any curve whose projection on the plane of XZ is given, is easily found. Let ABCD be the cylindrical area required. 162 INTEGRAL CALCULUS. [AitT. 143. Let y=fx (1) be the equation of AB, the line of iutersection of the surface with the phvue XY \ and let z = Fx (2) be the equation of CiT),, thi; projection of CD on the plane of XZ. If x,y,z are tlie coordinates of any point P of CZ>, the i-equired area is evidently the limit of the sum of rectangles, of which p pi pit put Jg ^^j^y Qj^Q -pjjg j^j.^,g of pp'pop'i' differs by an in- finitesimal of higher order than ds from zds, and therefore the required area -S" = | zds. x,z are the coordinates of Pi, and satisfy (2), and ds= 'sjdsr + djf where a*, // are the coordinates of F and satisfy (1). We have, then, r'z\ldx'-\-dy\ [3] For exami)lo, let AB be the quadrant of a circle, and let the projection of the required area on the plane of XZ be the quad- rant of an equal circle, so that the surface required is one-eighth of the surface of a groin. Here x- + y' = a\ (4) nd (5) ds-'^dx^ + di/'-'-^dx- ^'^^ , and -Ja'-; Chap. XL] AREAS OF SURFACES. 103 Therefore (Ir Again, let us find tlie area of the curved surface of tlie portion of a cylinder of revohition included within a splifrical surface, whose centre lies on the surface of tlie cylinder, and whose radius is equal to the diameter of the cylinder. If the centre of the sphere is taken as the oritrin, and a diametral plane of the cylinder as the plane of XZ, tlie surface required is four times that indicated in the figure. The equation of the cylinder is x~ — ax-\-f = 0, {C^) and of the sphere af + y- + z--a- = 0. (7) Subtract (6) from (7), and we get z- + ax-cr=0 (8) as the equation of a cylindrical surface perpendicnlar to the plane XZ, and passing through all the points of intersection of (6) and (7). (8) is, then, the equation of the projection on the plane of XZ of the line of intersection of the given spherical surface and the given cylindrical surface. From (G), From (8), Hence S ds = V(/.tr + (I if = — dx = " '^1/ ^^/ax-x" z = Va^ — ax. - I Va- — ax • aVn r"dx_ _ 2. adx __ ay a r" v a — x . dx 2 Ju VxVtt — X and the whole area requued, 4S = 4a\ 164 INTEGRAL CALCULUS. [Akt. 144. Examples. (1) Find the area cut from the cylindrical surface whose base in the phiue XI' is a quadrant of the curve x^ -\-yi = ai l)y the plane x = z. Ans. fcr. (2) Find the area of that portion of a cylindrical surface whose base in the plane of XY is a quadrant of the ellipse — _!-•— = 1, and whose projection on the plane of XZ is bounded r .1^' 2 ,0 o, 2 ^^ i e a6(rt-+rt6 + &') by the curve a^z- = 6-ar(cr — ar) . Ans. b = — ^^ -■ ^ ^ ^ 3(a + 6) (3) Let the base of the cylindrical surface be a tractrix, whose vertex lies at a distance a to the left of the origin, and whose asymptote is the axis of F, while its projection on the plane of XZ is bounded by the parabola z^ = —2'nix. Ans. S = 2a\/2ma. (4) Let the base of the cylindrical surface be the upper half of a cycloid, having its vertex at the origin and its base parallel to the axis of F, and at a distance 2 a from the origin, while its projection on the plane of XZ is bounded by the parabola z*=2mx. j_ns. S = 4:a^/am. Any Surface. 144. Let x, ?/, z be the coordinates of any point P of the sur- face, and X -f- Ax*, ?/ -f Ay, z + \z the coordinates of a second point Q infinitely near the first. Draw planes through P and Q parallel to the planes of XFand YZ. These planes will inter- cept a curved quadrilateral PQ on the surface ; its projection p7, a rectangle, on the plane of XZ ; and a parallelogram p'q^ not shown in the figure, on the tangent plane at P, of which ;>{a + x)\x .dx; Vrt r V.1 I snr\l .f?x = .rsin 'a I _ ./ \« + A- \((+.f 2 ./ a -\-x .dx. C'livr. XI.J ARKAS OF srUFACKS. 1G9 Let ii: = y/x ; 2 nxlw = dx r^Jx.dx rid-dw /V a \ -^ f' + -f V "^ V«/ I siu~\ — — dx Jo \a + x Z 4 4 2 8o- = 8a^ is the whole surrjice in question. 146. Let us find the area of the curved surface of a right cone whose base is the curve x^ + 2/* = a^ Jvud whose altitude is c. If we take the vertex of the cone as the origin of coordinates, and its axis as the axis of Z, the equation of its curved surface is x^ + f^^C^)\ (1) and the projection of the surface on the plane of Xi'is hounded by the curve xi + f, = ah (2) From (1) we get DJ \ a^ xlyi ' where x^y are the coordinates of any i)oint within tiie projec- tion of the base of the cone. Since the four faces of the cone are equal, the required surface 4 r- /^^«*-^^» , 170 INTEGRAL CALCULUS. [Art. UC. Let us substitute v^ = x aud tcr'^ = y, whence dx = 3v^dv and dy ='6io'dw, and we have o- = — I i vw Va^-y W -f- c^ (v^ + w'f . diudv ; fi c/O »/0 or, since in a definite integral it makes no difference what letters we use for the variables, a = — C Cxy y/cv'x'y'-\-c\x' + ijy . dydx. (4) The X and y in (1), however, must not bo confounded with the X and y in (3). The integral in (4) is precisely that which we should have to find if we sought the area of a surface of such a nature that its projection on the plane of XY was a quadrant of the circle ^.2 _|_ ^2 _ ^i^ jj^jj^i ^|jg secant of the angle made by the tangent plane at any point (x,y,z) of the surface with the plane of XT was xy \Jo? 3?y'^ + (?{a? + y-f. In the latter problem there is nothing to prevent our re- placing X and y in xy '\/o?s?y^ + c^ (x^ + y'^Y by their values in terms of r and <^, the polar coordinates of any point of the projection a.-^ + ?/^ = al, and dividing this projection into polar elements instead of rectangular elements, and then integrating between the limits which we should use if we were finding the area of the projection by the formula -^1=1 | rd(f>dr. We have, then, o- = — ( I r- sin <^ cos ^ Va^?** sin- cos- <^ + c;-?'* . rdrdA^ or jr Va^ sin^ cos^ ^ -h c^ . dr d^, «c/0 Jo i7=z C,(( i s'lucf) cos (f) Va^ sin''* cos'^ -{-c-. di{f) If the plane curve revolves aboyt the axis of F, ii^ Th^ volume of a si)]iere is \iTi^. l^fThc volume of the solid formed b}' the revolution of a cycloid a)*out its base is i)Tr-(i^. f40 The curve y'{'2a — x) = x^ revolves about its asymptote ; show tlmt-tlie volume generated is 2Tr-a^. (S) The curve xa +y^ = os revolves about the axis of X ; show that the volume generated is ^^'^^Trct^. Solids of Revolution. Double Integration. 151. If we suppose the area of the revolving curve broken up into infinitesimal rectangles as in Art. 137, the clement AajAy at an}' point P, whose coordinates are x and ?/, will generate a ring the volume of which will differ from ^-n-y^x^y by an amount which will be an infinitesimal of higher order than the second if we regard Ax and A?/ as of the first order. For the ring in question is obviously greater than a prism having the same cross-section A.rA_?/, and having an altitude equal to the inner circumference 2 Try of the ring, and is less than a prism having A.x-A.?/ for its base and 2ir{y -|- A?/), the outer circumfer- ence of the ring, for its altitude ; but these two prisms dilfer by 27rAa;(A_7/)'-, which is of the third order. Chap. XII.] VOLUMES. 177 'liriidy, where the ii^jper Hmit of iiite^r:itioii is the onli- uate of the point of the curve iinmediately ahove /', and nui.st be expressed in terms of .«• by the aid of the equation of the revolv- ing curve, will give us the elementary cylinder used in Art. 150. The whole volume re(|uired will be the limit of the sum of these cylinders ; that is, V=-2irr Cychjdx. [1] If the figure revolved is bounded by two curves, the required volume can be found by the formula just obtained, if the limits of integration are suitaljly chosen. Let us consider the following example : A paral)oloid of revolution has its axis coincident with the diameter of a sphere, and its vertex in the surface of the sphere ; recjuired the volume Ijctween the two surfaces. Let y- = 2mx (1) be the parabola, and ur + y- — 2 ax = (2) be the circle, which form the paraboloid and the sphere by their revolution. The abscissas of their points of intersection are U and 2 {a — in). We have V=2Tr j \ ydydx, and. in performing our fir.st integration, our limits must be the values of y obtained from equations (1) and (2). We get r = TT j [2 (a — vi)x — ar](?.r, and here our limits of integration are and 2(« — m). Hence F= |7r(a — m)^ = — r^ (j if h is the altitude of the solid in (piestion. EXAMIM-KS. ^l) A cone of revolution and a paralioloid <>f revoluliou have the same vertex and the same base ; recjuired the volume be- tween them. ,^^^. iTU^^ ^^.,^^,.^. ,^ .^ ^,,^. ,^^ii,„,,. „niH. cone. INTEfJKAL CALCCTLUS. [Art. 152. \2) Find the volume included between a right cone, whose vertical angle is 30°, laid a sphere of given radius touching it along a circle. . „ ttt' 6 Solids of Revolution. Polar Formula. l.')2. If we use polar coordinates, and suppose the revolving area broken up, as in Art. 138; into elements of which rd<{>ilr is the one at any point P whose coordinates are r and <^, the element rdcfidr will generate a ring whose volume will differ from 2 ir7~ sin (f>d(fidr b}' an infinitesimal of higher order than the second, if we regard d(f) and dr as of the first order ; for it will be less than a prism having for its base rd(f)dr, and for its alti- tude 2 TT (r-fdr) sin (<^ + d0), and gi-eater than a prism having the same base and the altitude 2 irr sin cf> ; and these prisms differ b}' an amount which is infinitesimal of higher order than the second. "We shall have then T'= 2 TT r fv-^ sin cf>drd, [1] the limits being so taken as to bring in the whole of the gener- ating area. For example ; let us find the volume generated by the r.^volu- tion of a cardioide about its axis. r = 2 a ( 1 — cos <^) is the equation of the cardioide ; 2 7r I I r»h\(f)drd<^. Our first integral must be taken between the limits ;• = and = 2 a ( 1 — cos fs\ud(t>. o F=— a^Trj (I — cos0)^sin/ Solid. Tn'jib' InU'ijrdfion. 153. If we suppose our soUv planes parallel to the three oo/irdinate planes, the elementary parallelopiped at any point (.r,.?/,2;) within the solid will have for its volume A.vA/yAri;, or, if we regard or, ?/, and z as inilependent, dxdydz ; and the whole volume V=ff^d,'dyaz, [1] tlie limits being so chosen as to embrace the whole solid. The integrations are independent, and may be performed iu any order if the limits are suitably chosen. As it is imjiortant to have a perfectly clear conception of the geometrical interpretation of each step in the process of linding 180 INTEGRAL CALCULUS. [Art. 153. a volume by a triple integration, we will consider one ease in detail. Let the integrations be performed in the order indicated by the formuh. r r C y=\ 1 ) dzdydx. If the limits are correctly chosen, our first integration gives us the volume of a pi'ism one of whose lateral edges passes through any chosen point P,(.'c,^,z) within the solid, is parallel to the axis of Z, and reaches directly across the solid from surface to surface, while the base of the prism is the rectangle dydx ; our second integration gives the volume of a right cylin- der whose base is a plane section of the solid, passes through the i)oint P, and is parallel to the plane FZ, and whose altitude is dx ; and our third integration gives the volume of the whole solid. The limits in our first integration are, then, the values of z belonging to the point in the lower bounding surface and the point in the upper bounding surface which have the coordinates X and y ; the limits in the second integration are the values of y belonging to the two points in the perimeter of the projection of the solid in the plane of XF which have the coordinate x\ and the limits in the third integration are the least value and the greatest value of x belonging to points on the perimeter of the projection of the solid on the plane of XK It is easily seen from what has just been said that the limits in the second and third integrations are precisely those we should use if we were finding the area of the projection of the solid by the formula ^ ^ A= \ I dydx. Of course, it is necessary to have a clear idea of the form of the solid whose volume is required. For example , let us find the volume of the portion of the ellipsoid ^ 2 2 cut off l)y tlie coordinale planes. Chai>. Xll.] VOLUMES. 181 and our limits are, for 2, and c\|l — ^— ^; Ibr >/. and I ? \ "" ^' 6-^1 — -^; and for x, and a. For, starting at anv point {.r.jf.z) and integrating on tlie liypotliesis that z alone varies, we get a column of our elementarv parallelopipeds having (ivdi/ as a base and passing through the point {x,y,z). To make this col- umn reach from the plane XY to the surface, z must increase from the value zero to the value belonging to the point on the surface of the ellipsoid which has the coordinates x and // ; that is, to the value c-^l — '—^—jo- Then, Integrating on the h}- pothesis that y alone A'aries, we shall sum these columns and shall get a slice of the solid passing through {xj/,z) and having the thickness dx. To make this slice reach completely across the solid, we must let y increase from the value zero to the greatest value it can have in the slice in question ; that is, to the value which is the ordinate of that point of the section of the ellipsoid b}- the plane XF which has the abscissa x. The section in question has the equation a' ^ b' therefore the required value of ?/ is ft \|1 — ^■ \ a- Last, in integrating on the hypothesis that x alone varies, we must choose our limits so as to include all the slices just Examples. UO Kind tlie vohune obtained in tlie present article, perlorni- 1- O' mg the integrations in the order indicated by tlie Ibrnuila, > V^C C Cdxdiidz. ^i Find the volnnie ent off from the surface z- V- C h by a planj^parallel to that of FZ, at a distance a from it. Ans. ird^-^ibc). Find the vohnne enclosed by the snrfiices, x' -\- y- = cz, or + y- = ax, 2 = 0. (4) Obtain the volume bounded l)y the surface z = (i — V.f- + y- and the i)lanes x = z and x = Ans. 32 c 2 a'' Chap. XII.] VOLUMES. jj^3 U^*>fY'\Vi^ the voliinie of the conoid ])ouii(Ie(l by the surface z^j^^ = (? and the planes .r = aiul x = a. Ans. ^. af 2 154. If we use polar coordinates we c^an take as our element of volume r^ sin ^r//Y/^(/^, an expression easily obtained from the element '2irrs'mody referred to a set of three roctang.1 ar axes fixed in the body, and let x^y^z be the coordi- nates of the centre of gi-avity. Place the body with the axes of X and Z horizontal, and consider the tendency of the particles of the body to i)roduce rotation about an axis through {x,y,z) paraUcl to OZ, under the influence of gravity. Represent the mass of an elementary parallelopiped at anv point {x,y,z) b3- dm. The force exerted by gravity on dm is measured bv dm, and i™t!!r.,';V"'r"°" ■' 'f "'■ "■ ""■ "■•""' <"■ '"» «•«■« ™"»n- tiatcd at 1 , the moraeut o^ tUc force exeitol on dm about the CllAl'. XIII.] CENTRES OF (JUAVITY. 185 axis through C would be {x — jc)dm, anil this monicut would rei)resent the tendency of dm to rotate about the axis in (jues- tion ; the tendency of the whole body to rotate about this axis would be 1{x — x)dm. If now we decrease dm indefinitely, the error committed in assuming that the mass of dm is concentrated at P decreases indefinitely, and we shall have as the true expres- sion for the tendency of the whole body to rotate about the axis through C, I {x — x)dm ; but this must be zero. Hence I (x — x)din = 0, I xdni — ^i dm = 0, xdm f^ J dm [1] If we place the body so that the axes of Y and X are liori- zojital, the same reasoning will give us jyd7 I din and in like manner we can get I zdvi ^=^4 [3] I dm Since (dm is the mass of tiie whole body, if we rei)resent it by Mwe shall have ^ I xdm y = ' M M I zdm 186 INTEGRAL CALCULUS. [Art. 157. / Example. Show that the effect of gravity in m.aking a body tend to rotate about any given axis is precisely the same as if the mass of the body were concentrated at its centre of gravity. 157. The mass of any homogeneous body is the product of its vohune l)y its density. If the body is not homogeneous, the density at an}- point will be a function of tlie position of that point. Let us represent it by k. Then we may regard (bn as equal to kcIv if dv is the element of volume, and we shall have / xkcIu [1] I Kdv and corresponding formulas for f/ and z. If the body considered is homogeneous, k is constant, and we shall have I xdv I xdv I ?/^J^ I ydv I zdv I zdv ■'="j^='— ' f^^ * V f^3 In any particular problem we have only to express dv in terms of the coordinates. PUmo Area. 158. If we use rectangular coordinates, and are dealing with a plane area, where tiie weight is uniformly distributed, we have dr = dA=dxdy. (Art. 1.36). Chap. XIII.] CENTIIES OF (IKAVITY. Hence, by i:)7, [2] and [r,], Oxdxdi/ l/ _ ffydxdy jjaxay If we nse polar coordinates, dv = fL-1 = rd ) ^ cos 0r?<^ i r?-2 d<^ 2 a^ n 1 - cos (^)- fZ<^ j (cos . Find the centre of gravity of the curve y- = h-^!~^- Ans. x = \a. Iml the centre of gravity of the cycloid. . y^ xhis. x = aTr, y = |a. //^ Find the centre of gravity- of the lemniscate r = (rcos2<^. \y^" A - 7rV2 Ans. x = a. H Find the centre of gravity of a circular sector. Ans. If we take the radius bisecting the sector as the axis a sin a of X, and rg^^resent the angle of the sector by 2a, x = f ind the centre of gravity of the seirment of an ellipse cut h drantal chord. Ans. a; = f // = S the centre of gravity of a quadrant of the area of the y-h = al. Ans. X v = m.- 159. If we are dealing with a homogeneous solid formed by the revolution of a plane curve about the axis of X, we have (Iv = 'Iirydydx. (Art. i:.i [1]; Hence, by Art. 157 [2], j jxijdxdy ffydxdy [1] -."V' \ OH ,0 \ 190 INTECUAL CALCULUS. [AuT. 159. If we use i)ol:ir coordinates, dv = '2Tn-sin. (Art. ir,2 [1].) I I ?'''sin(/)Cos<^(?rf7y the sHrface z- = xy and the five planes a;=0,,2/=0, z=0^ x=a, i/=h. Ans. a-='"|«, y=fb, 2 = ^'j«i/y4. 160. If we are dealing with the arc of a plane curve, tlie formulas of Art. 167 reduce to I ads x = '^, [1] ■r , ' I yds fi. Examples. l/i. Find the centre of gravity of an arc of a circle, taking the diameter bisecting the arc as the axis of X and the centre as tiie origin. iJir Find the centre of gravity of tlie arc of the curve x'i-\-y^=n\ between two successive cusps. Ans. x = y= j((. 3. Find the centre of gravity of the arc of a semi-cycloid. .1//.S. .r = (7r-^)*(, »/==-$ a. 4. Find the centre of gravity of the arc of a catenary cut off by any horizontal chord. Ans.x = 0, y = -— — -^, where 2.S is the lentrth of the arc. 2 .s- />f Obtain formulas for the centre of gravity of a surface of revolution, the weight being uniformly distributed over the surftice. INTEGRAL CALCULUS. [Art. 161 6. Find the centre ol' gravity of uny zone of a sphere. Ans. The centre of gra\ity bisects the hne joining the centres of the basgf of the zone. A cardioide revolves about its axis ; find the centre of gravity of tjje surface generated. Ans. x = — Ya** ('- ind the centre of gravity of the surface of a lieniisphero when the density at each point of the surface varies as its per- pendicular distance from the base of the hemisphere. ^^ Ans. x = ^((. ^. Find the centre of gravity of a quadrant of a circle, the density at any point of which varies as the vfh jiower of its distance from the centre. ^,jg jj; _ -^ _ ^' +'2 2a ■ ' ■ 7t + 3" tt' \^. Find the centre of gravity of a hemisphere, the density of which varies as the distance from the centre of the sphere. Ans. x = |a. Properfies of C'lildin. 161. I. If a plane area revolve about an axis external to itself through any assigned angle, the volume of the solid gene- rated will be equal to a prism whose base is the revolving area and whose altitude is the length of the path described by tlie centre of gravity of the area. II. If the arc of a plane curve revolve about an external axis in its own plane through any assigned angle, tlie area of th.y A rt . 1 58 [ 1 ] . C Cdxibi CiiAl'. XIII.] CKNTUKS OF CItAVlTY. ll»3 The loniitli of the path th-scrihcd hy the centre of gruvity (") I I ydxdy The voUiine gem-vated is y= ^-^ I i >/<^->^di/, by Alt. 151. Hence V=T/@i i dxdij. But I I dxd)/ is the revolving area, and the first tlieorem is established. "We leave the proof of the second theorem to the student. EXAMPttS. *}-rTind the surface and volume of a sphere, regarding it as generated by the revolution of a semicircle, a ^ -^^ y^. Find the surface and volume of the solid generated by the revolution of a cycloid about its base. l^i\¥'\nd the volume and the surface of the ring generati'd by the revolution of a circle about an external axis, Ann. V=2iT-a-h. S^Airab, where h is the distance of the centre of the circle from the axis. lAf^nd the volume of the ring generated by tiie revolution of an ellipse about an external axis. Ans. V=27r\dj(:, where c is the distance of the centre of the ellipse from the axis. /h^ -1 /v r" ^ "f^ \ ^^-r 194 IJS'TEGUAL CALCULUS. [Art. WL LINE, SURFACE, AND SPACE INTEGRALS. 162. Any variable which depends for its value solely upon the position of a point, as, for example, any function of the rectangular or polar coordinates of the point, may be called a point- function. A point-function is said to be continuous along a given line if its value changes continuously as the point, on whose position the function depends for its value, moves along the line ; it is said to be continuous over a given surface if its value changes continuously as the point is made to move at pleasure over the surface ; and it is said to be continuous throughout a given space if its value changes continuously as the point is made to move about at pleasure within the space. 163. If a given line is divided in any way into infinitesimal elements, and the length of each element is multiplied by the value a given point-function, which is continuous along the line, has at some point within the element, the limit approached by the sum of these products as each element is indefinitely de- creased, is called the line integral of the given function along the line in question. If a given surface is divided in any way into infinitesimal elements such that the distance between the two most widely separated points within each element is infinitesimal, and the area of each element is multiplied by the value a given point- function, which is continuous over tlie surface, has at some point within the element, the limit approached by the sum of tliese products as each element is indefinitely decreased, is called the surface integral of the given function over the surface in question. CiiAi'. XIV.] L.IXE, SURFACE, STAGE INTKCiUALS. 1%') If a given space is divided in :iuy way into infinitesimal elements such that the distance between the two most widely separated points within each element is infinitesimal, and ti»e volume of each element is multiplied by the value a given point- function, which is continuous throughout the space, has at some point within the element, the limit approached by the sum of these products as each element is indefinitely decreased, is called the sjxice integnd of the given function throughout the space in question. It is easily seen that the line integral of unity along a given line is the length of the line ; that the surface integral of unity over a given surface is the area of the surface ; and that the space integral of unity throughout a given space is the volume of the space. In the chapter on Centres of Gravity we have had numerous simple examples of line, surface, and space integrals. 1G4. That the value of a line, surface, or space integral is independent of the position in each element of the point at which the value of the given function is taken can be proved as follows : The distance apart of any two points in the same infinitesimal element is infinitesimal (Art. 163), therefore the values of a continuous function taken at any two points in the same element will differ in general by an infinitesimal ; the products obtained by multiplying these two values by the mag- nitude of the element will, then, differ by an infinitesimal of higher order than that of the element ; therefore, in forming the integral either of these products may l)e used in place of the other witliout changing the result. (I. Art. KJl.) 165. The line integral of a function along a given line is absolutely independent of the maimer in whicii the line is broken up into infinitesimal elements, and is equal to the K-ngtli of the line multiplied by the mean value of the function along the line; the vieaii value of the function being defined as fol- lows: Suppose a set of points uniformly distriluited along the 196 1^TKGKAL CALCULUS. [Akt. 165. line, that is, so (listiil)iited that the mimbor of points iu any portion of the line is proportional to tlie length of the portion ; take the value of the function at each of these points ; divide the sum of these values by the number of tlie points ; and the limit approached by this quotient as the number of the points is indefinitely increased is the 7nean value of the given function along tlie line ; and this mean value is in general finite and determinate. To prove our proi)ositiou, we have only to consider iu detail the method of finding the mean value in question. Let the number of points in a unit of lengtli of the line be k. Then, no matter how the line is broken up into infinitesimal elements, the numl)er of points in each element is A; times the lengtii of the element. Since any two values of the function corresponding to points iu the same element differ by an infinitesimal, in finding our limit we may replace all values corresi)onding to points in the same element by any one ; hence the sum of the vaUies cor- responding to points in the same element may be replaced by one value multiplied by the number of points taken iu that element,* that is, this sum may be replaced by k times the product of one value by the length of the element ; and the sum of the values corresponding to all the points taken in the line may be replaced by k times the sum of the terms obtained by multiplying the length of each element by the value of the function at some point within the element. When we divide this sum by tlie whole number of points considered, that is, by k times the length of the line, the A;'s cancel out, and the required mean value reduces to the limit of the numerator divided by the length of the line, and the limit of tiie numerator is the line integral of the func- tion along the line. Therefore the line integral is the mean value of the function multiplied by the length of the line. The same proof may l»e given for a surface integral or for a space integral. The former is the product of the area of the surface by the mean value of the function over the surface ; the latter is tiie volume of the space multiplied by the mean value of the function throughout the space ; and both are inde- Chap. XIV.] LINE, SUKFACE, SPACE INTECIIALS. lit? pendent of the way in winch the surface or si)ace may he divided into intiuitesimal elements. 166. If the line along which the integral is taken is a plane cui-ve, it is easy to get a geometrical representation of the integral. For, if at every point of the line a perpendicular to the plane of the line is erected whose length is equal to the value of the function at the point, the line integral n-quired clearly represents the area of the cylindrical surface containing the perpendiculars if the values are all of the same sign, and represents the difference of the areas of the portions of the cylindrical surface which lie on opposite sides of the line if the values of the function are not all of the same sign. A similar construction shows that a surface integral over a plane surface may be represented by a volume or by the differ- ences of volumes. Consequently, in each case if the function is finite and continuous, the integral is finite and determinate. 167. As examples of line, surface, and space integrals, we will calculate a few moments of inertia. The moment of inertia of a body about a given axis may be defined as the space integral of the i)rodiK"t of tlie density at any point of the body by the square of the distance of the point from the axis; the integral being taken throughout the space occupied by the body. If the body considered is a material surface or a material line, the integral reduces to a surface integral or to a line integral. In the examples taken below the Ixxly is supposed to be homogeneous. («) The moment of inertia of a circumference alK)ut a given diameter. Using polar coordinates and taking the diameter as our axis, / = I a- sin" (/) • k(i(i^ = ka^n = iMa\ [1] 198' INTEGRAL CALCULUS. [Art. 167. if I is the moment of inertia, and a the radius, k the density, and M the mass of the circumference in question. {h) The moment of inertia of the perimeter of a square about an axis passing through the centre of the square and parallel to a side. 7=2 Cy-My + 2 Cc^Mx = |i^/«^ [2] if '2 a is the length of a side. (c) The moment of inertia of a circle about a diameter. Xa rt2TT I r sin- (f> . knl({>dr = ^ k-n-a* = iMa\ [3] (d) The moment of inertia of a square about an axis through the centre of the square and parallel to a side. /= I I y^kdxdy=^ka* = iMaK [4] (p) The moment of inertia of the surface of a sphere about a diameter. I a^ sin" ^ . A;a- sin (jidcftdO = ^ kira* ./o = ^MaK [5] (/) The moment of inertia of the surface of a cube about an axis parallel to an edge and passing through the centre. 7=1 f" C\a- + z'')kdxdz -J- 2 f" C\y^ + z^)kdydz = "j' ka' + \£ ka* = ^iMa\ [6] Chap. XIV.] LINE, SrUFACE, SPACE INTEGRALS. IW {g) The moment of inertia of ii si)liero alxnit a diameter. 1= i i ( ■»" siu- ^ . kr siu drd(ie = y\ ^-Tra^ = iMcr. [7] (h) The moment of inertia of a cube about an axis through the ceutre and parallel to au edge. / = £" f" C\y^ + z^)MxcJych = -'/A-'a* = |3/a^ [8] Examples. Find tlie moments of inertia of the following bodies : ) Of a straight line about a jierpondicular through an extremity ; about a perpendicular through its middle point. y^ Ans. iMl'; ^\Ml\ i^z) Of the circumference of a circle about au axis through its centpe^perpendiciilar to its plane. A)is. Mo?. JJ(P) Of a circle about an axis through its centre perpendicular to its nlttne. . Ans. ^Ma?. L<4) Of througli its ^A) Of a rectangle whose sides are 2 a, 26, about an axis through its centre perpendicular to its plane ; about au axis through its centre parallel to the side 2h. Ans. i3/(«- + 6-); \Ma\ Of an ellipse about its major axis ; about its minor axis ; about an^axis through the centre peipendicular to the plane of theejKl^e. Ans. \MU'\ \Mcr\ \ M {(.r -{- U') . ^^6) Of an ellipsoid about the axis a. Ans. I .V(//- -f- r). ]J(f) 0f a rectangular parallelopiped about an axis through the c^trc parallel to the edge 2«. Ans. \M{U' -\-r). \/8) Of a segment of a ]Kira])ola about tlie principal axis. Ans. i-V6-, where 2 6 is the breadtii of the segment. H- 200 INTEGRAL CALCULUS. [Akt. H>8. 168. If u, D^u, and DyU are finite, continuous, and single- valued for all points in a given ^^Zarte surface hounded by a closed curve T, the surface integral ofDy^xx taken over the surface is equal to the line integral of ncosa taken around the li'hole bounding curve, where a is the angle made with the axis of X by the external normal at any point of the boundary. This may be Ibniiulated thus : C CD^udxd;/ = Cu cos a. ds. [1] Let the axes be chosen so that the surface in question lies in the first quadrant, and divide the projection of T on the axis of 1'' into infinitesimal elements of which any one is dy. On each of these elements as a base erect a rectangle ; and since T is a closed ciirve, each of these rectangles will cut it an even number of times. Let us call the values of u at the points where the lower side of any one of these rectangles cuts T, «,, u.,, »3, u^, etc., re- spectively ; tlie angles wliich this side makes with the exterior normals at these points, u,, a^,, u;,, a^, etc. ; and the elements which the rectangle cuts from T, rf.s,, (Zsji ds^, da^, etc. It is evident that whenever a line parallel to the axis of X cuts into the surface bounded by T, the corresponding value of a is obtuse and its cosine negative ; that whenever it cuts out, CliAV. XIV.] LINE, SURFACE, SI'ACE INTEGKALS. 201 a is acute and its cosine positive ; aud tliat any value of a 18 the angle which the contour T itself makes at the point in ques- tion with the axis of Y if we suppose the contour traced by a point moving so as to keep the bounded surface always on the left hand. We have then approximately, dll= —dSi •COSui = f/.s'o- C()Sa._,= —ds.^- COSuj = f?S4- C0Sa4='". [2] If, now, in | | D^udxdy we perform the integration with respect to .r, and introduce the proper limits, we shall have j \D, ndxd;/ = Cd;/ {— n^ + u.,— //,.j + »4 • • • ) ; [3] and the second member indicates that we are to form a quantity corresponding to that in parenthesis for every rectangle which cuts T, to multiply it by the base of the rectangle, and then to take the limit of the sura of the results as all the bases are nidefinitely decreased. By [2], f??/ (—?/ 1 -I- ?^, — ?<..;+».,••• ) = Vi cos ai dSi 4- H., cos a^ r/.s\, + ii.. COS ttg dS:^ + "4 <^'t>« "1 ''•"''i + • • • ; [-^^l and the limit of the sum of the values any one of which is represented by the second member of [4] is clearly | aconads taken around the whole of T. F^XAMI'I-K. iPp^Tve that under the conditions stutiMJ in ilic last article C Cj)^Ndxd>/= Cncosfi.ds, where /3 is the angle made with the axis of 1' liy tlie exti-rior normal. 202 INTEGRAL CALCULUS. [Akt. 169. 169. As an illustration of the last proposition, let us find the centre of gravity of a semicircle. We have ^ ^ lu ffy^^^^- ( ^ ) But we may write y = D^{xi/). Hence, by Art. 1G8, y = -^ffy^^-''^^!^ = 17 J -^'^ cos ads = — { i acos^asin^cos . ds = ki a cos (}> a- shr(;oscf> ad (f> = Aa^ f sin> cos-dcf> = - Tra^ = -Ma% Jo 4-1 which agrees with the result of (*•), Art. 167, EXAMI'LKS. ^^) Find the centre of gravity of a semicircle, using the theorem | i D^udxdy = i ucosfS .ds. Y/(2) Find the moment of inertia of a circle about an axis through its centre perpendicular to its plane, using the principles I I D^udxdy = J v cos a . ds and | | DytaLxdy = | ?< cos/? . ds. Chap. XIV.] LINE, SURFACE, SPACE INTEGRALS. 208 170. Since, as we have seen in Art. 168, a is the angle which the curve T makes with the axis of Y; if we trace the curve so as to keep the bounded space on our left, it follows that cosa.cZs = dy. Hence | | Djidxdy = I udy ; [1] and in like manner, JfD,ndxd>j = -f>alx; [2] the first integral in [1] and [2] being taken over the bounded surface, aiid the second around the bounding curve. For example, the moment of inertia of a square about au axis through the centre and parallel to a side is I=kC Cfdxdij. (((?) Art. 1G7.) •^ '- -• ' \ { y-'f-i-'^U = j xyhhj, and the last integral is to be taken around the perimeter. Hence l/o Example. York Ex. 8, Art. 1G7, by the aid of (2). 171. If U, D„Tj, DyU, and 1)^U are finite, continuous^ single-valued functions throughout the space bounded by a given closed surface T, the space integral of D,U taken throughout the space in question is equal to the surface integral, taken over the bounding surface, of U cosa, where a is the angle made with the axis of X by the exterior normal at any point of the surface. This may be formulated thus : r r f />. Udxdydz = Cl Vos a . dS. [ 1 ] 204 INTEGRAL CALCULUS. [Art. 17L The proof is almost identical with that given in Art. 168, except that for elementary rectangle we use elementary prism. We shall merely indicate the steps. dydz = — rf*S'i costti = dS., cosog = — dS._, cosag = ... r r f/A Udxdydz = Cfdydz [- b\ + U,-U,--] = the limit of the sum of terms of the form Ui costti . dr.AIULITY. 2U7 the quotient obtained by dividing the line integral of r takoii around the circumference by the length of the cu-ciiinfiTL-nce ; that is, I rds 'lira The polar equation of the circle is r='2a cos ; ds= 2a(Ict>, M= -L ( '4 a- cos <}ct> = — , '2iraJ T TT the required mean value. {h) Let us find the mean distance of points on the surface of a circle from a fixed point on the circumference. Here, by Art. I60, the required mean is the surface integral of r taken over the circle, divided by the area of the circle ; that is, 2.. cos* ^,^^ (c) The problem of finding the mean distance of points on the surface of a square from a corner of the S(iuare can be sim- plified slightly by considering merely one of the halves into which the square is divided by a diagonal. Here ^ Z ..»«'■<:'<• M= -„ I r- rdrd4> a-J» J" = |(V-i-Mo.t„niL'). 208 INTECniAL CALCULUS. [Art. 17', ((/) As im exumple of a (U'vit-o often oiiiployotl, we shall now solve the problem, To lliid the mean distance between two points within a given circle. If M be the required mean, the sum of the whole number of cases can be represented by (7rr)^iT/, r being the radius of the circle ; since for each position of the first point the number of positions of the second point is proportional to the area of the circle, and may be measured b}' that area ; and as the number of possible positions of the first point may also be measured by the area of the circle, the whole number of cases to be con- sidered is represented b}' the square of the area ; and the sum of all the distances to be considered must be the producit of the mean distance by the number. Let us see what change will be produced in this sum by in- creasing r hy the infinitesimal dr ; that is, let us find d{Trr*M). If the first point is anywhere on the annulus 2 Trr.dr, which we have just added, its mean distance from the other points of the circle is — , by (b). 9ir Therefore, the sum of the new distances to be considered, if the first point is on the annulus, is '-:^.7r7^.2irrdr; but the second point may be on the annulus, instead of the first ; so that to get the sum of all the new cases brought in by increasing r by dr, we must doul)le the value just obtained. Hence d{Tr"-r'M) = ip _,.v,,.^ TT-d'M = J-p TT Cr\h = Vb^-Tra*, 45 TT 17.">. In solving questions in Prohahilitii. we shall assume that the student is familiar with the elements of the theory as given in " Todhunter's Algeljra." (a) A man starts from the bank of a straight river, and walks till MooM ill a laiidoiii dirrction ; he then turns and walks CllAP. XV.] MEAN VALUE AND I'Kl UtAKI MTV. 2Ult in anotluM- random diivctioii ; wl>:iL is tlu' proliahility tliat In- will reach the river by night? Let ^ be the angle his Ih-st course makes with the river. If the angle through which lie turns at noon is less than - — -Id. he will reach the river by night. For any given value of 6, then, the required probabilit}- is ^ ~ " . The i)r()I)ability that 6 shall lie between an}' given value 6q and ^o + ^W is — . The chance that his first course shall make an angle witli the river between 6q and Oy^ + dO, and that he shall get back, is " TT-'ie (W {iT--26)d6 As 6 is equally likelv to have any value between and—, the required probability, (6) A floor is ruled with equidistant straight lines; a rod, shorter than the distance between the lines, is thrown at ran- dom on the floor ; to And the chance of its tailing on one ol" the lines. Let X be the distance of the centre of the rod frona the nearest line ; 6 the inclination of the rod to a perpendicular to the jjaral- lels passing through the centre of the rod ; 2a the connnon dis- tance of the parallels ; '2e the length of the rod. In order that the rod may cross a line, we must have ccos^ > x\ the chance of this for any given value Xo of x is . — cos ' -'. The prol)ability that x will have the value .r^ is — . The probabilit}- required is 2 C _,.r 2 c ttci^^o c -a This problem may be solved I»y another nutliod which pos- sesses considerable interest. 210 INTEGRAL CALCrLUS. [Art. 175 Since all values of a; from to a, and all values of 6 from — ^ to - are equall}' probable, the whole number of cases that can arise may be represented by I I dxdd = TTtt. The number of fa^'orable cases will be represented by I idxde = 2c. Hence j) = — TTCI (c) To find the probability that the distance of two stars, taken at random in the northern hemisphere, shall exceed 90°. Let a be the latitude of the first star. With the star as a pole, describe an arc of a great circle, dividing the hemisphere into two lunes ; the probability that the distance of the sec- ond star from the first will exceed 90° is the ratio of the lune not containing the first star to the hemisphere, and is equal to V2^ ~"'\ The probabilit}' that the latitude of the first star TT will be between a and a-\-da is the ratio of the area of the zone, whose bounding circles have the latitudes a and a -f- da respectively, to the area of the hemisphere, and is 2 ira^ cos a da , = cos tt tta. 2ira^ Hence p= i ' ^2^~ «) ^^^g ^ ^^^ _ _, ^0 TT TT (d) A random straiglit line meets a closed convex curve ; what is the probability tliat it will meet a second closed convex curve within the first? If an infinite num1)er of random linos be drawn in :i pl:ino, all directions :iie etpially prol)al)le ; and lines having any given Chap. XV.] MEAN VALUE AND TKOIiAHILITV. ill direetiou will be disposed witii ecjiial IVeqiieiiey all (ner the plane. If we determine a line by its distance p from the origin, and by the angle a which p makes with the axis of X, we can get all the lines to be considered by making p and a vary Itetween suitable limits b}- equal infinitesimal increments. In our problem, the whole number of lines meeting the exter- nal curve can be represented by j | dpda. If the origin is within the curve, the limits for }) must be zero, and the perpen- dicular distance from the origin to a tangent to tlie curve ; and for u must be zero and 2 it. If we call this number N, we shall have N= ipda, *^ p being now the perpendicular from the origin to the tangent. If we regard the distance from a given i)oint of any closed convex curve along the curve to the point of contact of a tan- gent, and then along the tangent to tlie foot of the perpendicu- lar let fall upon it from the origin, as a function of the a used above, its differential is easih' seen to be pda. If we sum these differentials from a = to a = 27r, we shall get the perimeter of the given curve. Hence N = I pda = Z/, where Jj is the perimeter of the curve in question. By the same reasoning, we can see that 7i, the numl)er of the random lines which meet the inner curve, is equal to ^ its perimeter. For p, the required probability, we shall have EXAMIM.F.S. !„^A number ?i is divided at raiidoin into two parts ; find the mean value of their product. 1,,^. '!^ " () * 212 INTHGUAL CALCULUS. [Akt. 175. Find tlio mean value of tlic ordinates of a semicircle, siip- p(5sing tlie>ierie.s of ordinates taken equidistant. ^,^5^ ^^^^ (P9 Find the mean value of the ordinates of a semicircle, sup- posing the ordinates drawn through equidistant points on the circumR'^uce. . 2(( (P) Find the mean values of the roots of the quadratic ar — ^'Si ) Suppose a straight line divided at random into three parts ; find the mean value of their product. , a^ y " '"• ¥)• 1/(7) Find the mean square of the distance of a point within a iiiven square (side = 2r/) from the centre of the square. •^8) A chord is drawn joining two points taken at random on a circumference ; lind tiie mean area of the less of the two seg- ments iato which it divides the circle. . ira- a- lr[U) Mud the mean latitude of all places north ot the etiuator. A71S. 32°. 7. 10) Find tlu^ mean distance of [)oiuts within a sphere from ven point of the surface. Ans. |a. (11) Find the mean distanc I wirfiin a sphere. •f l\v<> points taken at random Ans. 44 a. (12) Two points are taken at random in a given line a ; find Ihe chance that then- distance shall exceed a given value c. .■1//.S • tlie ilis- tance between consecutive lines. (20) A plane area is ruled with equidistant parallel straight lines, the distance between consecutive lines being r. A closed curve, having no singular points, whose greatest diameter is less 214 INTEGRAL CALCULUS. [Art. 175. than c, is thrown down on the area. Find the chance that the curve falls on one of the lines. Ans. ■ — , where / is the perimeter of the curve. TTC (21) During a heavy rain-storm, a circular pond is formed in a circular field. If a man undertakes to cross the field in the dark, what is the chance that he will walk into the pond? ClIAP. XVI.] ELLIPTIC INTEGKALS. 215 CHAPTER XVL ELLII'TIO INTEGRALS. 176. In attempting to solve completely the prol)lem of the motion of a simple pendulum by the methods of I. Ciiapter VIII. we encounter an integral of great importance which we have not yet considered. The problem is closely analogous to tliat of the Cycloidal pendulum (I. Art. 119). For the sake of simplicity we shall suppose the pendulum bob to start from the lowest point of its circular path with the initial velocity that would be acquired by a particle falling freely in a vacuum through the distance ?/o; and this by I. Art. 114 [1] is V2^o. Forming our diflferential equation of motion as in I. Art. 118, but taking the positive direction of the axis of Y upward, we ds Multiplying by 2 — and integrating, ds\ . , ^ .dt. or, determining C, v' = (~IJ= '2 (,{!,.-!,). (2) If the starting-point is taken as the origin, the equation of the circular path is ar + y- — 2 ay = 0, whence \dt) 2cvj-y\dtJ and we have , "^ , '^'= V2^o-y), V2 ay — y^ "' 216 INTEGRAL CALCULUS. [Art. 1/ dt= ^^ V2gr . N{y^-y){'lay-f) lutegrating, and determining the arbitrary constant, we get t = ^ c ^y — (3) ^'2gJ^ V(yo-2/)(2ai/-/) . as the time required to reach that point of the path which has the ordinate y. The substitution of .r- = — reduces (3) to the form ^^i"T7 dx ^((1 -.')(. -f;.-' w where the integral is of the form C ^^ — . (5) *^» V(l -ar')(l-fc2a^) fc^ being positive and less than unity if ?/u is less than 2 a. An examination of equation (2) will show that if this is true, the pendulum will oscilhite between tlie two points of the arc which have the ordinate ?/„. If ?/o is greater than 2 a, tlie pendulum will make* complete revolutions. For this case the substitution of x^ = f- in (3) will reduce it to t^aJ^f' "" (6) V<-<-ir?) where the integral is of tlie form (o), k- being positive and less than unity. The time required for the pi-ndiilum to reach its greatest height — that is, in the first case, the time of a half-vibration, and in the second case, the time of a half-revolution — will depend upon r ^•^- — . (7) Chap. XVI.] ELLIPTIC INTEGKALS. o-jy 177. TIio leiiiith of an art; of an Ellipse, measured from the extremity of the minor axis, has been found to be (Art. 107) XV^fcS-''- 0) If we replace ' by x, (1) becomes a and the integral is of the form IVt ^ • clx, (3) where k- is positive and less than unity. The length of an Elliptic quadrant depends upon the integral XV^ -^.dx. (4) 178. It can be shown l)y an elaborate investigation, for which we have not room, that the integral of any algebraic function, which is irrational through containing under the sciuare root sign an algebraic polynomial of the third or fourth degree, can by suitable transformations be made to depend upon one or more of the three integrals F{k,x) =r^=r^£^=-=, [1] Jo V(l-ar)(l -A:V) n{n,k,x)= r ^^ [31 ^' {\+ ux') V( 1 - .1-) (1 - A-»x») which are known as the Elliptic Integrals of the lirst, second, and third class respectively. 218 INTEGRAL CALCULUS. [Art. 179. A;, whicli may alwaj'S be taken positive and less than i, is called the 7nod(dus; and ?«, which may be taken real, is called the parameter of the integral. K= F(k, 1 ) = f ^^ [4] dx and E=E{k,\) =xvw-' [^^ are known as the Complete Elliptic Integrals of the first and second classes. 179. The substitution of iB = sin<^ in the Elliptic Integrals reduces them to the following simpler forms. F(A-,*)=r-^!^= =r|*. [1] n („, ic, '« (l+J«sin2<^)Vl-^--sin"<^ c/«(l + ?isin- [3] K= c _A^_ = r'i^. [4] Jo Vl-Fsin^,^ Jo A<^ ^ = r Vn^"Fsin^ .d<^ = f A . r7<^. [f)] <^ is called the amplitude of the Elliptic Integral, and A = Vl — A:'sin^ having the period tt. It has its maximum value 1 Ck.vp. XVI.] ELLII'TIC INTKOKALS. 219 when ^ = and when <^ = tt, and its niiuinuira vahie \' 1 — k\ which is usually represented by A' and called the complemenUu-y modnhis, when )=r-—^ Jo ^J^ _ z-2 , Vl — k-sm-cf) we replace <^ by <^i, , and l)eing connecicd by the relation , , sin2-^ tand) = ^-^—, (1) ^'-^cos2<^l '^ which is easily transformable into either of the following : ^•sin<^ = sin (2 ^1 — ^), (2) tau(<^-c^,) = f^-f tanc^,, (3) I ^ reduces to — - — | ' — ^0 Vl-^-^sin^<^ 1+^v^o L i^sin^c^, which is also an Elliptic Integral of the first class, but has a different modulus and a different amplitude from those of the given integral. The steps of the process are as follows : From (1) we easily find 8in2 2<^, siu-c^ 1 +^-^+2A:c()s2 _ secV 1 +A- + 2A-cos2<^i. (A; + cos2<^i)^ ' 2(l+keos2o 1 + A;- + 2 k cos 2 , 2d<^i 2d, Vl-A;^sin> Vl +A;2+ 2fc cos2<^i Vl + A,-^+ 2A;-4A;siu"'^0, rfc^i \-i-k x(l ^,sinVi ) = -^ F{k„cj>,). where ^•i _ 2VA-^ 1 + A-' and sin(2<^ -) = A siii<^ A-, is less th .111 1 and < greater th [4J ; for < 1 reduces 14- A; to 0<(1 — Va-)-, which is obviousl}' true, and > A; 1+A: reduces to 4 > A(l +A-)-, which is true, since A- is less than 1. If ^ is not greater than ^, and the smallest value of <^i con- sistent with the relation sin (2<^, — ^) = A' siu<^ is taken, < <^i < c^. Ilonce (4) is a reduction formula by which we can raise the nioduUis and lower the amplitude of our given function. CllAP. XVI.] ELLIPTIC intk(;i:als. By applying the fonnuljv (4) n times, we get 221 or, since l+A- 1-fA-, 1+A\ = — -, etc., -l-A-„ , nK,n)\ F{k, ) = A-,. ^^•^^•^-K . ^(^. _^ ^^ ) ^ where A- ^ 2VA; I 1 + K I and sin ( 2 <^^, — <^^,_i ) = A-^, i sin <^^,_, . D'] If we suppose ?i in (5) to be indefmitely increased, we shall have rA-,,] = 1 ; for if we form the series (l-A-) + (l-A-04-(l-A-,) + -"+(l-Av)-f--, we shall have 1 - Av 1 - A-, 1 - k/ 1 + va-^ 1 + k; which is always less than unity ; hence the terms in the seric-a must decrease indefinitely asj) is increased and _ [1 — A"„] = 0. Since, as we have seen above, <^„ continually diminishes as n increases, but does not reach the value zero, it nuist have some limiting value . Hence ^^-"^ ^^ Vl -sin-<^ = 1 sec (I(f> = log tan ^ + - ; and F(k, 0) = log tanf"^ + |1* /5SI^. [6] Formulas [/)] and [6] lend themselves very readily to numer- ical computation. 222 INTEGRAL CALCULUS. [Art. 181. 181. Formula [4J, Art. 180, may be used to decrease the modulus and increase the amplitude of a given Ellii)tic Integral. Interchanging the subscripts, and using (3) Art. 180 instead of (2) Art. 180, we have F{k,)='^-thF(k„cf>,), where and Jc, 1-vr 1 4.Vl-A;^ tan(<^, — (^) = Vl — k-tiin4 By repeated application of [1] we get [1] i^(A:,<^) = (l+A-,)(l+A-,)...(H-/0 ^(^•«, ..) when and tan(<^^ - <^^^i)= Vl - A--^ 1 tan<^^ 1. [^] limit It is easily shown, as in Art. 180, that _ [A'„] = 0, and limit /** consequently that _ F{k,„ <^„) =1 d(j!) = , where 4> is the n 00 ^y limiting value approached by (f> as n is' increased. If <^ = ^, we get from [2], c/>i = tt, . = 2 7r, ...<^„ = 2"-^; hence K=f(j.; ^) = ^ ( I + h) (1 + /.■.) (1 + A-.) [3] Formulas [2] and [3], like foiniulas [;">] and [6] of Art. 180, lend themselves readily to computation. "NVitli a large modulus, it is generally best to use [5] and [6] of Art. 180; with a small modulus, [2] or [3] of the present article will gonorally work more rapidly. We give in the next article the whole work of computing the Elliptic Integral /'Y-^, - ) by each of the two methods, and Chap. XVI.] ELLIPTIC INTEGRALS. 223 of computing k(-—\ = f(—^,'^A by tlie second method, employing five-place logaritlims. 182. Ff^,-\ Metiiou of Art. 180. fc= 0.70712 log^ = ;».s4949 1+^-= 1.70712 log (1+ A;) =0.2^226 log V^ = 9.92474 log2 = 0.30103 colog(l+^')= 9-76774 logA-i = 9.99351 A"i = 0.98518 logA:i= 9.99351 l-i-A;i= 1.98518 log(l +Ai) = 0.29780 log VA?i= 9.99676 log2 = 0.30103 colog(l 4- A-i) = 9.70220 logA-,= 9.99999 h,= 1 logic = 9.84949 log sin ^ = 9.84949 2*^1 - log Va'i = 9.99676 cologV^ =0.07526 9.48277 log-F' colog/x = 0.36222 V2 2 4 = 9.91701 i?'/^:Zl e"\ = 0.82605 V 2 uy /ui = 0.43429 is the modulus of the common system of logarithms. rf^,"^ Method of Art. 181. VI -/.•- = /.' =0.70712 1-/,' =0.29288 1 +/.' = 1.70712 /.•, = 0.1 7157 1_7,-, =0.82843 1 +/.-, = 1.17157 Z-,'= 0.98520 1 -/.,'= 0.01480 1 +/./= 1.98520 A:2 = 0.00746 ,()g(l -/,-') = 9.46669 c'ol()g(l + A-') =, 9.76774 logA-, = 9.23443 log(l -/.•,) = !). 91826 lou(l + A:,) = 0.06878 log A-,'- = 9.98704 log 7.-,' =9.^9352 Iog(l -A/) = 8.1 7026 colog ( 1 + AV) = 9.70220 logA-2= 7.87246 Chap. XVI.] KLMI'TIC INTEGRALS. 225 1 _ h, = 0.992 ')4 log ( 1 - A-,,) = 9.9907") 1 + k., = 1 .00746 log ( 1 + /.-,.) = 0.003-23 log A-,'- = 9.99998 log A/ =9.99999 logy logA-' = 9.84949 log tan <^ = 0.00000 log tan ((/>! — <^) = 9.84949 ^-cf> = Sij° 15' 53" , = 80° 15' 53" logA.V = 9.99352 log tan ^1 = 0.76557 log tan (c^2 - <^,) = 0.75909 <^, _<^i= 80° 7' 17'' .,= 160° 2:3' 10" tan (<^3 — <^o)=tan^2 4) = <^3 = 2 <^, = ;520° 46' 20" 1$= 40° 5' 48" = 144348" TT = 648000" colog7r"=4.18842 log 77=0.49715 log(i*) =i 9.84499 226 INTEGKAL CALCULUS. [Aur. 183 log (1 +A-i) = 0.06878 log(l 4- A2) = 0.00323 log('|3^ = 9.84499 logi?'(^^,^') = 9.'J1700 F{^,'-^\ = 0.82G05 Vi For F[~,~\ we have by (3), Art. 181, log(l +A:i)= 0.06878 log(l +/t2) = 0.00323 log7r = 0.49715 colog 2 = 9.69897 logi^f— , 'M = 0.26813 i V2 1.8541 2 2, 183. Landen's Transformation can also be applied to the computation of Elliptic Integrals of the second class, but the task is a more dillicult one ; we shall, however, give a brief sketch of the method ; and in so doing we shall apply it to a more general form -^'J VI — A''sin-> of which S(fc, <^) is a special case. From Art. 180 we have . /i 'n~^ ^n 1 + A; cos 2 ,) ^ l + A;^+2Acos2 0i ^ Chai-. XVr.] ELLIPTIC INTKCltALS. 227 Hence Vl — k" sin- (f> + kcos ^ = Vl 4- k' + '2k cos 2 <^,. •^^'Vl —k'-sm-cji A. l-(i-k-i ^-2sin-> ^" Vl-A-sin-> J =r a + .Vl-A-sin2<^ /c" -#.vr= A:^sin-<^ rf<^, and G(k, <^)-^sin. A' ^0 [_V1 — /i-sin-^ A. Snbstitnting Vl+A;- + i cos 2 ) = ^sin c^ + -" <7,(A„ c^,), [2] where A-, = |^;, sin(2 0i-c^) = A-sinc^, a.= --f, y8i=\^- [••^] Formnlas [2] and [.")] enal)le ns to niaUe onr given fnnction depend npon one of tlie same form, bnt liaving a greater modulus and a less amplitude. A repeated use of [2], together with the reductions emi)loycd in Art. 180, gives us 228 INTEGRAL CALCULUS. [AllT. 183. G(k. <^) = |sin<^ + Asin<^,-f- ^1 • /8osin«^j + fcn ^MiJ^> . 6^,. ( A-„, <^„) , [4] where /8^= " ^ and /3A I 2 J 2^ ^ ^ 2^-^ V A' \^ ki /I'l A;^ A.'i ^^ A-'a • • • A^ ly Just as in Art. 180 k,^ rapidly approaches 1 as n is increased ; the limiting vahie of 6r„(A-„, <^„) is then limit G,XK, „)= (•"""■> + /^''^'">d<^ c/« cos d> COS(f> = (a„ + /3„) log tan (j + 1^) - i8„ sin ). [4] can thus be written Gik,^) = F{k,^)L-^J\+l+ "" flJi ATg ... A;„ ] k'l Ao A3 ... A'„ j/ J ^\ A-i A,As + ^ sin + ~ sin , + — ^ sin 2 +— ^^ sin <^ + ^L VA- VAA-, ^/kk,k2 2'i-i 2" "I , sin (^„ 1 - ,_., sin <^„ . [7] VfcA-i...A;„ 2 VaAv-Vi J CllAP. XVI.] ELLII'TIC INTKGUALS. 22'J If a = 1, and fS = — A-, [7] reduces to A-'i A*2 • • • A^rj- 1 A'l a'2 • • • a',j _ — A- sin ^ + -^ sin c^i H — ; — ^sin(^2 + "* 2 . 2- — sm(^i + ---^ Va- Va-a-, siu d),. VA-A'i...A-,._,, ^kki...k„ _ 2VA~ in<^«i [«] where A;^ = " /^' , and siu (2 <^;, - <^,, ,) = ^V i siu <^p i- [9] i + A-;, 1 By Formulas [8] and [9] an Elliptic Integral of the Second Class may be computed without ditliculty. 184. Formula [2], Art. 183, may be used to decrease the modulus and increase the amplitude of an Elliptic Integral. Interchanging the subscripts, we have G (A-, 4>) = ^-^[g, (A-„ <^0 - 1 sin <^,1 ; or, since f = f . (Art. 183 [3]), A"i 2 G where (A-, <^)= ^-4^'| G, (A„ c^,) - /^ sinc/,,1 [1] fc, = f7-'{L^, tan(<^,-<^) = Vr-A-='tan<^,«, = »+^,/?, = ^''/. [2] 230 INTEGRAL CALCULUS. [Akt. 184. A repeated use of [1] gives G-(A, n By Art. 181, [2], ^-±h . i±h...l±h>^^^ = F{k, , 4- ^^^^' siu «^, + • • •], [a] where i-^i-kU K = 'A -'^"^1 tan (<^^ - <^^, i) = Vl - A; i • tan ^ ,. 1 + Vl - A-; 1 [C] We have seen in Art. 181 that if ^ ='^, d, =9p i_ Therefore, for a complete Elli[)tic Integral of the second class we have Formnlas [o] and [7] are admirably adapted to compntation. We give in the next article the work of computing E[ — , -) by each of the methods jnst aiven, and of com- puting ^(~o"'o) ^^y the second method; using, as far as possible, tlie values already employed or oljtauied in Art. I.s2. 185. Ef'^,-\ Mf.tiioi) of Art. 183. Here, as we have seen iu Art. 182, if we carry the work only to five decimal places, k.,= 1, and our working formula will be E{k,cj>) = F(k,)\']+kfl-pj] r '* 2- "I — k\ sine/) -(-— ^siu(/), — ■ siuf/j. • L V^ Va-a-, i 232 INTEGRAL CALCULUS. [AuT. 185. log 2 logA- colog A'l 0.30103 9.84949 0.00G49 log(^ 0.15701 ^=1.43553 A:, 1+A:= 1.70712 1 + A- -^■'\ = 9.43391 9 ;• 1 + A; - — = 0.27159 ogF/'^,-^ = 9.91 701 9.35092 '&■ i)C+'-f)='-''''''' log A; =9.84949 log sin = 0.6 log2 = 0.30103 logV^' = 9.92474 log sin (^1 = 9.78445 0.01022 — sin x= 1.0233 VA; log 22 =0.60206 log V^= 9.92474 colog V77i = 0.00.324 log sin «/>2= 9.78122 2 - - sin VA: ^'^ sin<^o- 2.0477 VA;A-i 0.31126 - JcfsUKf) -\- ^ sin <^.,^ = 0.5239 V/Ta^ 7 ). :i^(^,; \ 2 log^-, = 9.23443 logA-2=7.'^7246 Colog4 = 9.39794 6.5U483 2- ^ = 0.00032 ^=0.08578 2 l+^ + M?=i.08610 1(1+1 + ^-^'') = 0-^71525 logO.728475 = 9.86241.5 1 _ |Yl + ^ + ^^^ = 0.72847o logFf^^,j) = 0.91 700 V-i 9.77941.0 FfJir,!!-] (0.72847.5)= 0.GU17H IogA-= 9.84949 logV^, = 9.61722 colog2 = 9.69897 log sine/., = 9.99370 9.1.5938 A;A- <>, = 0.14434 234 INTEGRAL CALCULUS. [Art. 185. logfc = 9.84949 logV5 = 9.61722 log VA;2 = 8.93623 colog4 = 9.39794 log sin ^ = 0. 14646 i?'/'2^, '^Vo. 728475)= 0.60178 Ef^, -\ = 0.7482i Ef^,'!^\ Method of Art. 184. :^'2)=^t^'i)[^-iO^^'^o} l--fl +h^hh\ = 0.728475 logO.728475 = 9.862415 2 V -^ '^' / WF/^— .-^ = 0.26813 :'2J ° V 2 2 CllAi'. XVI.] ELLTPTrc INTEGRALS. 235 186. An Elliptic Integral of the tirst or second class, whose amplitude is greater than -, tan he made to depend upon one whose amplitude is less than '^, and upon the corresponding Complete Elliptic Integral. AVe have F{7c,.)= f^= r^+ f^ = A'+ i'\ bv [4], Art. 179. In r^ let <^ = TT - i/. ; then d(}> = —dij/ and A<^ = Vl — Jc' sin^<^ = Vl — A"- sin'-'i// = Ai/', n !: and we have r d^ ^ _ T # ^ _ rO^ _ r-d^ ^ j^ 2 2 2 Hence ^(., .)=p|^= 2/f. [1] F{k,n^ + p)=:J'^'^ A<^ ^fr?<^ /-j^ rd± ^ r±,„, f# ./ A(^ J A<^ J A(^ J A<^ J A " TT in- pn n" (p+nf In r^ let =i>TT + ip\ then (/<^ = (/ ^ , A^ 236 INTEGRAL CALCULUS. [Aut. 186. The substitution of if/ for <^ — ?i7r iu j — gives us nT+p p p J A(f> J A(// J Ad) Therefore i^(A;, mr + p) = 2?i/ir+ F(^-, p). [2] In like mauner it can be proved that F{k,ii7r-p) = -2nK-F{k, p), [3] E (A-, 7i7r + p) = 2 n^ + E (A-, p) , [4] E{k,n7r-p)=2nE-E{l\ p), [5] where E = E(k,-\ is the complete Elliptic Integral of the second class. A table giving the values of the Elliptic Integrals of the first and second classes for values of the amplitude between and - is, then, a complete table. Such a table, carried out to ten decimal places, is given by Legendre in his " Traite des Fonctions EUiptiques." We give in the next article a small three-place table. It must be noted that the first column gives F(0, ) and £(0, <^), that is, I d), that is, log tan /''' + -Vnd sin<^. Tiie complete Elliptic Integrals, /f=F(/.-.; „n,lA'=A7/,-, are given in the last line of eacii table. CiiAi'. XVI.] ELLIPTIC INTEGRALS. 237 t>i 4- -4 'p'pp C\ (^ in •-q C^ '-" O H- tsJ O NJ tJi ooo 4- bo N3 U)4- Cs pop •^J CO p Cn ^I P OOO ^icoo '-n -a O a, ^ 5' II ClO 3 II -^ U> to U>4^ ON OOO ooo -J ^ ^I O O tsi pc>p 4- u> to _ - 4- VI On OsO Oi— OJ pPP Ln ^1 p i-OO O vO CO oI^4^ poo PPP ^ODp VT^ P K- N) U) 2. i>r- 9 II coo loco 'COCO oi^i 00^ < -f' o< CO U\ ( OvDCO U) to to -t^COON 'OO OO 4^ 00 PPP ^ Cv '^ to U> OJ •^ OON -I c^ •-" Oj Oj Oj OOO PPP 4»- U) to 4» i^ On U> Ui U> 000 4>. U> to 4-!^. ON 000 000 t-op *J CO p U)P I— <^> CT> "^ VI H- t/3 I ;;n VI -i- 4., H- 1— • I W U) O oo< ^o< ^ico( B II !feO £9 oO 5r 238 INTEGRAL CALCULUS. [Art. 187. II e OO o odd OO — odd O t^'t- 8 CO t^ O- ddo Q CO l^ OO — odd 0\ N ro CO ro ■*- o t^ •^ o o — t^ t^ CO odd d> ONO ddd i^oco (M — ON W-j O NO ddd ON-:^CO I^NO 1- t^ CO C7N CO NO "^ ddd >/^ CN] ON CO t^ >o I^COOn ddd CO t^NO t^OOON ddd ro t^ O I ro — NO( tT cm . O — < \0 ) ; F(k, x)= r ^^^ .A. V(l -x-){l-k''x^) F(k.^)=r ''^ - =r^. Jo ^1 —k'^sm^^ -'" A) , we have in Art. 179 called ff> the amplitude of n, and sin0, cos<^, and A<^ may be called the sine, the cosine, and the delta of the amplitude of u ; and (/>, sin^, cos<^, and A^ may be written amw, sinam», cosamvt, and Aamw, or, more briefly, sxmu, sn?/, cn?<, and dun ; and may be read amplitude?*, sine amplitude ?«, cosine amplitudes, and delta amplitude it. Formulating, we have u = F{k,x) = F{k,), ^ = am?t, a; = sin ^ = sn ?t, )■ ^ [1] Vl —x^= cos <^ = en ?<, Vl-Ar^a;'=A<^ = dnw, Cll.vr. XVI.] ELLirXrC INTEflKALS. 247 suK, cnu, duM, are trigouotuetric fuuclious of <^, the ampli- tude of w, but the}' may be regarded as new aud somewhat complicated fuuctious of u itself, aud from this point of view they are called Elliptic Functions of u. amn also is sometimes called an Elliptic Function ; and there are various allied functions that are sonK-times included under the general title of Elliptic Fuuctious. We shall, however, restrict the name to sn », cu ;/, and dn «. They have an analogy with trigonometric functions, and have a theory which closely resembles that of trigonometric functions, and which we shall proceed to develop. It must, however, be kept in mind that the independent variable n is not an angle, as in the case of the trigonometric fuuctious. Of course, with our notation, u^F{J:, .r) =: sn~ ' (./;, k), or u = F (k, ) = am- • (<^, k). The fundamental formulas connecting the Elliptic Functions of a single quantity follow iunuediatcly from the defiuilious [1], and are su'?? + cn-(/ = 1, [2] dn-'» + /t-su^< = l, [3] [4] cl am u da = du», clsnu dn = en '/ . ( n u d en 11 ^ — sn n .di 1 u. da rfdn« [5] [6] = — A"sn«.cuM, [7] da The only one of this set which needs any explanation is [H 248 hence and, finally, Since we see that INTEGRAL CALCULUS. [Art. 103, A0 dnu d am II (In — (ln». Jo A<^ Jo A(— ^) Jo A^' am( — ?/) = — am?(, sn {— ii) = — sn », en (— u) =cnn, (In (— ti) =dnt;, [8] That is, sn?< is an odd function of u, and enw and dnu are even functions of w. Since we have X A<^ 0, am(0) = 0, sn(0) =0, cn(0) = 1, dn(()) = 1, J [9] 193. Our addition fonnula for the sine amplitude flows immediately from [;">], Art. 190. Let u= F{k,x) and v = F{k,y), and take the sine amplitude of each member of [5], Art. 190; we get sn (n -\-v) = snn . cn^ . dn^ + cnu . snv . dnu 1 — /i^ . sn-« . sn^v If now we replace v by — r, and simplify by [8], Art. 192, wc have , s snw . en?' . dn?; — cnw . snu . duM sn('f — r) = — — , and the two formulas can be com])ined if we use the sitrn ± ; Chap XVI.] ELLIPTIC IXTKGRALS. 249 y , ,^ _ sn u . cu i^ . du V ± en ic.snv . dn u p., 1 —Jir. sn-(( . sir-u From [1], with the aid of [2] and [;3], Art. 192, wo can get, after a rather ehiborate reducliou, tiie aildition foiuuilas for en and du. / j_ \ enw . cnv ^F sn?( . sn-y . diiM . dnv p,, 1 —k^ . sn-» . sn-y *• -^ , , . . dn?< . dur T i"^ . sn»< . snv . cn« . cnu tot dn (u ± I') = f^ ,^ _ [3] From [1]. [2], and [3J a huge iiumlier of formuhis can he readily obtained. AVe give only those for sn ; there are similar ones for en and dn. .... . x 2 sn It . en V . dn V r-.-, sn(?< + t') + sn(«- y) = -, ^ -• [4J 1 — ^- . sn- ?f . sn^y , , X / X 2cnw.snv.dnw r-i sn (u + v) — sn (u — v) = — — • [ol ^ ^ ^ ^ 1 -A;-.sir«.8n2y ■■ "^ sn(M + v).sn(w-'y) =- — • [0] 1 — AT • sn- II • sn'* /• , , / , \ / \ cn^v + sn-u . du'y r-n 1 — A"' . sn'^M . sn'^v 1 + k- sn ( H + V) . sn ( u - r) = — — T:; j -. [8] 1 — A- . sn*?/ . sn^y n I / I \n n 1 / \-i (cn r + sn '/ . dn c)-' ^oi From [2] and [.'5] conies tlic uscfnl fonniihi cu(» + v) = cnu . en I' — sn" . sn *• . dn ('/ -\-v). [10] 250 INTEGRAL CALCULUS. [Art. 194. 194. If in formulas [1], [2J, and [3] of Art. 193 we let v = w, we get the following formulas for sn 2m, en 2m, and dn2M: n 2sn?t . en?< . dn?t p-t sn2u = — — — ; , [1] 1 — k-sn*u I'u — SIT 71 . du^?A 1 — 2su'w + 7c^sn*u 1 — A;-sn'*w 1 — ^-sn^ [2] , n dn^M — A;- . sn-?( . cn-u 1 — 2k^sn^u + ]c^sn*tt ro-, >"''" T^:^¥^u — = — r^¥^u — M From those come readily , ,1 2sn-it.dn^M r>n 1— cn2«t = — - — , [41 l-dn2« = -- —— , [6] 1 — Arsn*M •• I 1 > 2dn^?< PTT l + dn2u = — — — . [7] 195. Replacing u by , and dividing [4] by [7] and [G] by [5], Art. 194, we have nu 1— cnit 1 — dnw 2 l+dn?t A;^(l+cnM) [1] jU _ dnM+ en» _ — Jc' ^ + Jc^ en u -{- dm i r-oT ^^2" l + duM ~ A-(l+cutO '' *- -^ , 2U _ k'^ + dnM + k'^cnu _ (en u 4- dn u) pg-i '^ 2~ 1 +dn?t ~ (1 +vnii) ' ' ^ ^ where A'^=l— ^'j and is the square of the complementary modulus. r iiAP. XVI.] ELLIPTIC IXTECRALS. 2ol From [1], [2], and [13], we can get without diniculty the set o» dn?< — enu p-t 2 k'- + dmc — k-cnu 2 ^'-' + dn«-fc"^cuw •- -■ ^^2Tf^_^^^(l+dn«)_, j-g-j 2 k'^ + dmi — k-cnu Numerous additional formuhis can be obtained by the exer- cise of a little ingenuity, but we have given the most useful and important ones, and they form a set as complete as the usual collections of trigonometric formulas. Periodkiti/ of the Elliptic Functions. 196. We have seen (Art. 18G, [2]) that F{k,n7r+p) = 2nK+F{k,p), [1] where A" is the complete Elliptic Integral of the first class. Let ?< = F{k, p), and take the amplitude of each member of [1] ; we get am ( ?/ + 2 )t K ) = n tt + am (/ ; [2] or, replacing n by 2?j, am ( " + 1 n K) = 2 ?j TT + am u ; [3] whence sn {it 4-1 /( A') = sn »/, ^ en (« 4--i» A') = cu's I; [4] dn (?t + 4?j A') = du'/, j and sn */, cii »/, dii u are periodic fimctions. and have the real period 1 A', dn '/ actually has tlu' smaller period 2 A'. :is mny be seen Ity taking the delta of both members of [2] 252 LNTEGKAL CALCULUS. [Art. li.^. Since the amplitude of A" is -, we have [5] and our addition formulas [1], [2], [3], Art. 193, give us readily (?< + A) = -— , dnrt en (w + A") = — ^■' sn u dun du(»+Ar) = dnu [6] sn(?t + 2A') = - ~snn, cn(?< + 2A')=: -cn?A, dn(« + 2A') = dUK, sn(?< + .'Ur) = - cn?( dn H cn(w + 37r) = k' sn V dn u dn(u-|-;]A') = dn 11 [7] [«] sn {u -\-4 K) = sn ?/, ^ en (71 + 1 A") = en '^ l , dn(?/ + 4 A') = dn?^, J [9] a confirmation of [4]. ^ .1'. XA^I.] ELLIPTIC INTEGRALS. 258 197. Jt is easy to get formulas for the sn, cm. :iinl dii of an imaginary variable, wV— 1, by the aid of a titinsfonnatioii thi'i to Jacobi. Let v = F{k,ct>)=r^^,, (1) so that <^ = a)nr, sin^^sne, and cos<^ = fnr. In (1). re- place = Vl + A^-'tan^, (4) and (J(f)= ^f — \ . seciA . dip. Since il/ and 4> equal zero together, Ju Vl-A-'-sin^'^ If now we let n = F{k', ij/) , we have -y = ?t V — 1 . (5) Hence, since li/ = ain » (mod//) , we have from (2), (3), and (4), sn(r, /.•) = V-1 — ) — 77(' ^ ^ en («, A- ) en (v, k) = — ; — —i ^ ^ en (w, A;') dn («, A:') ^ ■^ en (m, /c ) or, as i' = ?< V— 1, 264 INTEGRAL CALCULUS. sn (u, Ji') on («, k') sn {u V- 1, A-) = V- 1 en {u V— 1, /.-) = en (//, k') [Art. 197. m dn (u V— 1, A;) = ; jr: ' It is interesting to note tliiit if w i.s replaced in (6) by mV— 1, the formulas reduce to sn (— ii) = — sn?^ en (— n) = cwn, dn (—?;) = dn?<, and are still true. Consequently, in (G), u may he eitlier a real or a pure imaginary. Let rfl// ^ (mod/i') =./: d^ = K'. Vl-A;'-sin-> Then, by Art. 19G, 4/r' i.s a period for sn?t (mod/i'), en u (mod 7i') , and dn « (mod A') . Hence sn (ic\/— l + 4?i7t'' V— 1) = sn(( V— 1, en (u V— 1 + 4 II /r' V — 1 ) = en n V— 1 , dn (a V^nr+ 4 i> K' y/^\) = dnn V^^ ; or, replacing «V— 1 by r, sn {v + 4 /( A'' V — 1 ) = sn ?", en (r + 4 ;; K' V^l ) =^ en r, dn {v + 4 ?i 7r' V^n") = ^1» ^'^ and 4 /iT'V— 1 is a period for sn, en, and dn. We see, then, that our Elliptic Functions, like Trigonometric Functions, have a real period, and, like Exponential Functions, Lave a pure imaginary period. They are, then, what may be called ['] Chap. XVI.] ELLIPTIC INTKGKALS. 255 Doubly Periodic Functions, and they are often studied from the point of view of their double periodicity. Like Trigonometric Functions, the E^Uiptic Functions may be developed in series, and from these series their values may be computed, and tables resembling Trigonometric tal)les may be prepared. A partial three-place table is here [)resented as a sample. It is complete for Elliptic Functions having the modulus ~ ; that is, 0.7. Modulus — = 0.7. 2 n snu cni< dnu 0.00 0.000 1.000 1.000 0.05 0.051 0.9 We have E{k, 4>)= i A<^ . defy. Let u = F(k, (f)) , then eft = am u, and E (A-, <^) may be written E{k, amu), or, more simply, E{amu), if the modulus can be omitted without danger of confusion. Xani II du({ . d avail ; or, since by (4), Art. l'J2, damn = du^; . du, E (am?t)= I dn-)'.f?u. [1] As an example of the usefulness of the form just given in [1], we will employ it in getting an addition formula for Elliptic Integrals of the second class. E{&mu) + E (i^mv) = I dn-n.7 l\ei)laciiig z by u +z, and iviueinln'ring that u and v are given constants, I dn'-2 .dz= I dn- {u-\-z) dz, and E{amii)+ E(iiu\v) = E [am ((( + f)] - pr.'ln- ( n + z) -dn-^] dz. (2) dn-(M + 2;)— dn-2;= [dn {n -fs) + dn^] [dn (;< +2;) — duz]. (:i) We can obtain from [3], Art. lt)o, the foUowing formuhi.s analogous to [4J and [5], Art. 193, 1 / , \ , 1 / \ 2 dn ?( . dn v ... dnO< + iO + tlii(H-'y)=^ -^^; —, (4 1 — Arsu-u . sn-v , , , . 1 . X 2A,-^sn?t . snv. cnw . cnv 1 — A-sn-»( . sn-v If in (4) and (.">) we let ?t + v=(t+2;, and u — v = z, and substitute tiie results in (3), we get dn^ {ii + z) — (\\YZ 4 A-- su ( - + 2 ) en ( ^ + ^ (In - + 2; J sn _^ en ^ dn - 1 —k-su-'^sn-r^ + zj and I [dn- (u +z) — dn-z] f/2 = — 2sn c-n dn I 7= ^ ■: r^. „ u u , u 2sn on dn- 2 2 2 1 su'-- 1 — Jc- an- 2 U .,(\L , \ 258 INTEGUAL CALCULUS. [Art. 1'.)9. since - 2 Jc" sir" sn /^" + A <-n f''^ + z\ dii /"'* + A (?2 is the differ- enti 1 of 1 — A- sir- su- ( - + 2 V 2 V-' J - CM - au - •' '' H ! I "1 ■- 1 — /.- sir _ sir 7+1^ 1 — A- sn* - sn A-2sn-cn-dn- su- +y — sn''- 2 2 2 V2 / 2 1 _ A-- sn* - 1 - A-- sn^ " su^ f " + v 2 2 V2 = — k- . sn 11 . sn r . sn (?< + i') ■, by (i), Art. 194, and [(>], Art. 193. Hence by (2), E (nmu) + £'(ami') = E[am(ii-{- r)] + ^"snit . snv . sn(if + t'), our required addition foruuihi. APPLICATIONS. Rectification of the Lemniscate. 199. From the polar equation of the Lemniscate, 7-^= a^ cos 2^, referred to its centre as orijj;in and its axis as axis, we get as the length of tiie are, measured from the vertex to any point, P, whose coordinates are r and 6. s = a \ == a I — . [1 J -^" Vcos2^ -^^ Vl-2siir^ CiiAP. XVI.J KiJJi'Tic inti:(;i;ai.s. 2r)0 and foi- the arc* of tlu' quadrant of the Li-nmiscate, that is, the aic from vi-rti'X to conliL', ^^- c-^] ^" VI These diflfer from Klli[)tie Integrals of the first chiss only in that the coefficient of sin-^ is greater than nnity, and tliov may be reduced to the standard form by a simple device. Introduce in [1] <^ in place of 6, <(> and 6 being connected by the relation sin-d) = 2 s'urO. I'lieu we have Vl- 2 sin-/^ = cos<^, J2_ and ^^^^V2 cos<^fZ<^ Hence s =^^ C -^^— ^^^f^^.A •2 ^' Vl-^sin^'ci -2 V 2 V [••'] ayJ-2 C'__d±__ ^ aV2 ^AV2 7r\ * |- , j 2 -''^ ^/^^±'sin^^ -2 V 2 ' 2/ and .. . Vl — 4 sin-0 The auxiliary angle cfi is very easily constructed when the point P of the Lemniscate is given. We have r = aVcos2^, and we have seen that Vcos '26 = cos. For OQ = a cos AOQ and OP = a Vcos 2 6. 260 INTEGRAL CALCULUS. [Art. !<.»;). EXAMI'LKS. (1) Find the nunierieiil value of i (2) Kediicc (' ^^"^ to nn Elliptic Integral of tl Jo Vl-4sin- 1. jitis J— ( ^ whore sin' (//= n &\\\- <^. (3) Tlie half-axis of a Lemniscate is 2. What is the length of the aic of a quadrant? of the arc from the vertex to the point whose polar angle is 30°? Ans. 2.622 ; 1.168. In the inverse problem of cutting off an arc of given length the Elliptic Functions are of service. As an interesting example, let us find the point which bisects the quadrantal arc of the Lemniscate. , tiui-. and the corresponding point, P, is found by the method described on page 255. Rectification of the Ellipse. 200. We have seen in Art. 177 that the length of an arc of an Ellipse measured from the end of the minor axis is ■'x-2 dz. [1] If we let X = aaluff), [1] becomes s = a (''' V 1 ^^siu- . di> = a E (e, ^ ), [2] e, the modulus of the Elliptic Integral, being the eccentricity of the Ellipse. If x = a, <^ = '^, and the length of the KUiptic quadrant is n s^ = (, ("V 1 -^'='sin'^ . = a E (e, -\ [3] 2(32 INTEGRAL CALCULUS. [Art. 201. The length of an arc of the Elliptic quadrant, not measured from the extremity of the minor axis, can of course be ex- pressed as the difference between two Elliptic Integrals of the second class. The amplitude <;^, corres|)onding to a given point P, of the Ellipse, is easily constructed as follows : On the major axis as diameter describe a circumference ; extend the ordinate of P until it meets the circumfei'ence, and join the point of intersection with the centre of the ellipse. The angle the joining line makes with the minor axis is seen to be the required amplitude <^. If is given, P may be found by reversing the order of the steps of the construction. EIXAMI'I.KS. r- ?/- '" The equation of an ellipse is ^ — |- i- = 1 required the length of the quadrantal arc ; of the arc whose extremities liave the abscissas 2 and 2V2. ^l)(.s. ,").4 ; O.ltl 1. (2) Find the abscissa of tiie end of the- unit arc measnri-d from tiie extremity of the minor axis in the ellipse ^— + --- = 1 ; of the point which bisects the arc of the quadrant. Ans. O.OyC; 2.57. 201. \\y the aid of the addition formula E{-MnH) + E{i\\nv)=E [am (n+ w)] +7i;-snusu v&n{n+v) ([G], Art. 198) it is always possible to find an arc of an ellipse differing from the sum of two given arcs by an exi)ression which is algebraic in terms of the abscissas of the extremities of the three arcs. Tliis will bi! clearer if we modify slightly the form of our addi- tion formula. Ciivr. XVr.] KI.LIl'TK" INTKfMtALS. 263 Let c/) = :im», i// = tun /", and fr = aiii('/ + r). Thou tho forimila given above becomes E{h\ cfi) + E{k; 4,) = E{h\ o-) + A-- sine/) sin i/zsino-, [1] where ^, ip, and , ^) = aE (c, a) + —;x\ . x., . x^, if a'l, iCo, and x.^ are the abscissas of the points whose amplitudes are (f>, \f/, and a. The mos^ interesting case is when it = , in which case aE{e, cr) is the arc of a cpiadiant. [2] tlien reduces to = cos<^ cost// — sin<^ siui/^ V 1 — e^, or - siu)—\aE (e^'^\—aE{e, \p) = (f*'- sin<^ siin/'. [1] If, then, any point, 7*, is given, [.'^] will enable us to get the amplitude of a second point, Q, and thus to find Q, Q and P being so re- lated that the arc Bl\ minus tl»e arc AQ, shall be equal to a (piantity which is proportional to the product of the ab- scissas of /' and Q. 264 INTEGRAL CALCULUS. [Art. 202. For the special case wliero ^ and ip arc equal we have from V^l tan ^ = la and from m BP -AP = ■■ (le- sin = = a- -b This point, which divides the quadrant into two arcs whose difference is equal to the difference between the semi-axes, has a number of curious properties, and is known as Fagyiani's point. Examples. (1) Show that the distance of the normal at F^agnani's point, \ from the centre of the ellipse, is equal to a — h. (^ y (2) Show that the angle between the normals at P and Q, in . N the figure is equal to i// — ^ ; tiiat tlie normals are equidistant *^ from ; that this distance is BP — AQ. Rectification of the Hyperbola. 202. If the arc of the Hyperbola is measured from the vertex to any given point, P, whose coordinates are x and y, its length is easily found to be ■\dy, [1] b- or =s:-7r> ^^^ • +i. / if e is the eccentricity of tiie Hyperbola. Let ae b- ae . , Chap. XVI.] ELLIPTIC INTEGRALS. 265 and [2] becomes S = I ^= ; aeJo I 1 . 2 J ^1 --sm^ hence . = ^ r^^^^A^±^ = !l T «M^!!i^ r-n aeJo Vl-A;-sin2 — Jr c os- <^ A = tan ; ./o A but A-^^iirj^ = -L_A<^ A A<^. [4] The angle ^ corresponding to a given point P is easily con- structed. AVe have only to erect a perpendicular to the trans- verse axis at a distance — = from the origin ; that is, ae Vet- + b- at a distance from the centre ecpial to the projection of b on the asymptote, and to join the projection of P on this line with the centre. The angle made by the joining line with the trans- verse axis is A<^ is the distance from the centre to the normal at P. (3) Show that the limiting value approached by the difiference between the arc and the portion of the asymptote cut off by a perpendicular upon it from P, as P recedes indefinitely from Cuw. XVI.] ELLITTir INTKGUALS. 2i\l ferred to as the (Ufforenee between the length of tiio infinite :irc of the hyperbohi and the length of the asymptote. Show tliat in example (1 i this difference is ecpial to 2.803. The Poirlubim. 203. We have seen in Art. 17G that if a pcndulnm starts from rest at a point of its arc whose distance above tlie lowest point is i/o, the time reqnired in rising from the lowest point to a point whose distance above the lowest point is //, is 9^'" Vl-^•'sin-> where k = and sin d> = \|- — . mill siii the highest point reached l»y tlie pt-ndiilinn, and /' tiie point reached at the expiration of tlic time t. Call A(JB a, and AOP 0. Then ^ = 1 - cos«. and J-'^ = VA ( I -cos«) = sii>" = A". (I \2a " 2 Consequently the modulus of the Kllii)tic Integral in [1] is 268 INTEGRAL CALCULUS. [Art. 203. the sine of one-fourth the angle tlirougli which tlie penduhim swings. y= 1 -cos^, a and and ^2 a Vi(i — cos^) = sin - it Y . e \ 2 « 2 1 '/(I • a \2a 2 and therefore the sine of the amplitude of the Elliptic Integral iu [1] is easily computed when the angle through which the pendulum has risen is given. When 6 = a, sin^=l, and C/.: ; so that the time of a half-oscillation is i^i^it)' a confirmation of [7], Art. 17(!. The construction indicated in the figui'c gives the angle . AO'Q : ^T^ BUI J_ i • (1 -cos^0'g) = sin^^^ = sin.4CQ. Chap. XVI.] ELLIPTIC INTE(JUALS. 2n9 It is voiy easy to express the angle in terms of t. 1(1 ,-,/ . n We have ' ~ V -^'^1 ^^'^^ v» 2a the pendulum will make complete revolutions, and that the time required to pass from the lowest point to any point whose distance above the lowest point is y, is ,=„ )Z r- ^^^ =«,g ,,.(,,»), \ (l!h -/o Vl - A;- sin- \ .V.'A, where k = \\~ and sind)= \ •'— Show that in this ease fb= , and tiiat sin = sn [ - \ •^'" V ^2' 2 ^a\ 2 j Note. — In workiii<^ with a pendulum it is often about as easy to compute F (/.-, ) by developing by the binomial theorem and integrating two or three terms, as to use a table of Elliptic Integrals. f(,,,^)=r^?i — > Vi - , AV'e have A;-sin^^ ( 1 - 1c- sin- c-«)s<^) J» Vl — /i-sin-> -i 3 I) — ^ — A'sin''<^ cosc^ -f [—A' ( — sin (i> cosc^) •••• 32 01 CiiAP. XVI.] ELLIPTIC intk(;rals. 271 Differentiation and Integration. 204. Kewriting formulas [4], [5], [G], and [7], of Art. 192, wc have d am X = cln x dx, [ 1 ] (/ sn a; =: en a* dn x dx, [2] d Gnx = — sn X dn x dx, [3] d dn x = — k^ sn a; en a: dx, [4 ] we add o? tn a: = — 5- (Zx. fol cn'^x "- -■ By the usual method of differentiating an inverse function (I, Art. 72) we get readily d sn-i (x, k) = --==^= , [6] d en-^ (a-, A-) = , — , [ ( 1 d dn-i (a-, A-) = -=J^==, [8] dx d tn- » (x, A) = , • [9] ^ ^ V(l+;r=')(l + /.-'^a:») '^ "' [6], [7], [8], and [9] give at once a very valuable set of formu- las for integration, namely : V(l - x') (1 - /c'x') k'x^ : en-' (a-, k) = F(k, eos-'.r), [11] n dx ^- ^r(i-x^)(k'^-\- r'-=^_ = .ln-Vx,A-) = sn-'filEZ,/A 272 INTEGRAL CALCULUS. [Art. 204. dx f^ , — = tii-^ {X, h) = F (A-,tan-ia-). [13] If in [10], [11], [12], and [13] we substitute y = x^ and then change y to x, we get ("'-=^ ^— ^2sn-^(V^,A-)=2i^(A-,sin-'V!^),[14] Jo ^x(l-x)(l-k-'x) ^- ^/x(l-x)(k" + /rx) f , '^"^ - 2 du-i (V^, /.•) '^^ Vx (1 -x){x — k'^) = 2 F(^k, sin-' ^^^^^y [16] r , ^"^ = 2 tn-i ( Vx, 70 ^0 ■\/x{l+x){l + k"x) = 2F{k,tan-^\/x). [17] The following formulas are obtained from formulas [10]- [13] by easy substitutions : r-.=^== = isn-Y^^-\-->.>/.>0;[19] a>lj>x>0; [20] .r>/^>0; [21] r° ^ -^^ =ldn-Y^, iZEZ'Y ^^'^ y/(aF—x^(x^ — P) « V« « y a>x>I>>0', [22] f^ -^- - S-Y-- -^ZEi^Y [23] CiiAi-. XVI.] ELLIPTIC LNTEGllALS. 273 For example we will take [19]. Let y = - ' then dx^= '- • ^ ir Let now ,~ = tn/ and Jo V(i-ay)/ u' \ r- (ijl _i r^ (iz From [l-iJ-ClT] may be derived in like manner -*' V(x-a)(x-/3)(.f-y) Va-y V^-'— 7 ^""r, ,r>a>^>y; [24] f ''■' =-!=..-{ SE1.SE}\ ^^ ^(a-j:)(x-l3)(x-y) Va-y V^a-/S ^a-y/ a>^>/3; [IT.] ^y ^/(n-j')(ft-j-)(,— y) Va-y \^/i-y ^«-y/ y3>./->y; [I'O] ^' -J(a-.r){fS-.r)(y-J-) Va-y V^^--/- ^a-y/ y>-'-; [-''] For example we will take [24]. Let ?/ = » tlien dx ^ ^ ; x — y if 274 INTEGRAL CALCULUS. [Aux. 204. X -J{x-a){x-p)ix-y) d,, ^0 V//(l-(a y)y)(i-(/^-y)//) Let now z = {a — y)y, then Jo Jjj n — /'^ — .v^ / Vy(l-(a-y)y)(l-(^-y)2/) 1 r-^ 1 — ,, »^ii / dz Va — y From [24]-[27] may be obtained s:- V(x - a) (a; - )8) (x - y) (^ - 8) 2 V(a-y)(^-8) a->a; [28] V(a — X) (j (x-/3)(x-y)(:r-8) 2 / /^__8 „_^ Jg-^ y-A V^a-/3 CC-8 ^a-y fi-hj V(a-y)(^-S) a>r>y3; [29] Jx ■^(a-x){p-x){x-y){. (^-8) 2 sn" J\a-j,_^-x\p-y^a-^\ V(a-y)(/8-8) /3>.r>y; [30] CiiAi- \vi.] ELLIPTIC inti-:(;i:al8. 9 = sn- Y J^Zli . yjZ:!-, J— /^ . y - A. -8) \^y-8 )8-^- ^a-y /i-dj V(a-y)(/3 y>./->8; [;!1] dx X V(a-.r)(y3-r)(y-a')(8--T) V(a — y)(^— 8) \>a — 8 y - .>• >' a - y /^-8 8>... [;i2] Formulas [i^4]-[32] enable us to integrate the reciprocal of the square root of any cubic or biquadratic whicli has real roots. As an example let us find I =• *^*J V(2 ax — x^) (a- — xr) ,I.r V (2 (I — X) ('( — X) X (x + (i) V(2 a — x) (a — X ) .'• I -'• 4- //> (Ix - r"-= •^a V(2 a — a-) (« — a-) a- (j + ") =.l[sn-{l,f)-sn-(^4^)]bym _ 1 sn-/'^, ^^ - 1 //^,8ia->:^>\ 276 INTEGRAL CALCULUS. [Art. 204. Formulas [10]-[32] suggest the appropriate substitution to rationalize any rational function of x and the square root of a cubic or biquadratic having real roots. For instance, let us consider I V(«" — x-) {b- — x'^) dx. Let y = sn"'( y, - j, [v. formula [1<^]]. Then a; = ^ sn ( //, j , dx =^b en y dn // dy, iv^ — x^=^ a^ dn^ y Ir — ./■- = 1/ cu- y ; V(«^ — X') (U- — X-) . dx = (d>H en- y dn- y dy Examples. (1) Find r ,J^— • Am. ^ 7vYmod ^ V^ 1-311- (2) Kationalize J Vl — x* . dx. A71S. 2 V2 jT '((In^o- - dn*x) dx. fmod ^\ X'' dx VC«^ - hx) ( V(«' — Ox) {hx — x^) 2 / /A 2 / h\ Ins. - sn"M 1, - or -A' mod - • (4) Rationalize J J'l—Z]!!, dx. Avs. 2 substitute and reduce and J'** ax /- r' V^+ 0-2) ~" ^ V_ 1 V(l + //- Ki-y^) -2V2 f ^ -^- -2cn-r0,f ) Jo V(l + a;^) (1 - a:^) \ 2 ^ = 2 A' r mod ^ J =3.708. Examples. SiKjfjestlon : let s =: x^. (2) Rationalize f '— =^^=. -^" V.r(l+a;2) f'''l —en?/ Jo 1 + en // -^ r^'2-2cn?/ — sn'^?/ , / , V2\ = I ■-. • dif mod — - • Jo sn^y ''\ 2 J (3) Rationalize f \l I -\- x* ■ dx. Jo (1 +CI1 y)2 "^ Jo sn^r \ 2 / 20G. Formulas for integrating sna*, en a-, dna-, and their powers positive and negative, are obtained without difficulty. Chap. XVI.] ELLIPTIC INTEGRALS. 279 /I /•— A-sna-cna-r/j- 1 C (li( s\\xdjc = — —, I = — T I — A" J en X k J yj ,j-i _ /.'» = -l log (y + vy' - Z^-") if Z/ = tin or. Hence I su xdx = — - log (dn a- + Vdn- j — A-"*) = -ic„s,..(iiH). [,J I en .r dx = - cos~ ^ (du x). [2] I dii X dx = am a; = sin~^ (sn a*). [3] /dx _ /• sn cr en a- dn a- dx j P dij sn a" »/ sn'^ x en a* dn a: "J U'^J (1 — u) (1 k'^ii) =_ , log r V(i-y)(i-A-V)+i _ L±i:1 if y = sn^ X. Hence Jtoi: = 2l^'^"' L di?^ J 1 ^ , rA;'sna- — on./"] = — tan-M t; i • [6] A;' LA-'sna- + c-n.rJ •- -• From Art. 198 [1] we get f'sn^ X dx = ]., [.r - E (am .r. A-)], [7] 280 INTEGRAL CALCULUS. [Art. 206. cn2 X dx = - \_E (am x, k) — A-'^x], [8] / dn^ xdx=^ E (am x, k) . [9 ] An important set of reduction formulas by which the integral of any whole power of sn x, en x, or dn x can be made to depend upon the formulas just obtained can be found with- out difiiculty. We have — - (sn"'+ ' .r en x dn x) dx ' = {m + 1) ^iV'x—{m -f 2) (1 + A;2) sxr + '-x-\-{m + 3) 7r sn"' + -' a-, whence we get (m. + 1)1 ^n"" xdx = {m + 2) (1 + Z-2)J' ^^^-^ + ^xdx — (m + 3) k'-J sn'" + ■• x dx -f sn'" + ' a; en a: dn x. [10] (m + 1) k' ■ j en'" X dx = {in + 2) (/.•' =^ - Z-^) J" en'" + '-xdx + (m + 3) /.-^J cn"' + ^a- r/.r — cn'" + i a- sn a; dn x. [11] (/M +1} /.'- 1 dn'"a;(/a3 = (»i + 2) (1 + k' -) J' dn'" + 2.r rZx — (/« -I- 3) J dn'« + ■• X dx + /.•- dn'" + ' a- sn a; en x. [12] CiiAi-. XVI.] ELLIPTIC INTEGRALS. 281 Examples. (1) Obtain the following formulas : I sn ~ ' X (/x ^ x sn ~ ' a- + 7 cosh "" ' ( / ' ) j cn-» xdx = xcn-^x — - cos"' (V/.-'^ + A-^x^ j dn~^xf/x= a;dn~'x — sin~M ^ — ^ )• (2) Yim\ab£ fl-^^^i^'sn^x + ^'sn^rlr/x. ^n5. i ^ 1^0^^ + b') E [j^, 1^ - (a-^ + 2 //) A' (muA ^^^ V2 C" f V2\ (3) Find ~ J (dn^ x - du^ x) dx, i mod -^ j • V2 / V2\ ^ri5. ^/if mod ^y- j. or 0.211) ^. - r"2-2cna--sn='a: , / , V2\ Find / -, dx, I mod -— • Ji) sn^a; \ 2 / A'— 2 A' r mod ^ j- or O.aj;? (4) (5) A cannon-ball of radius h is fired horizontally through the middle of a ship's mast (radius a) ; find (a) the volume, and (b) the whole superficial area of the plug required to fill tlie hole. Ans. (a) •'^[(- + /'V^('' ")-^'^ + 2/,^,A(mud^^)]; (b) 8(a + /.;^'A('^ !^)-("-/')A'(n..Ml''^jJ- 282 INTEGRAL CALCULUS. [Akt. 206. (6) A cylindrical hole of radius h is bored through a sphere of radius a and just grazes the centre ; find (a) the area of the inner surface of tlie hole, (b) the s}3herical surface removed, and (c) the spherical volume removed. Ans. (a) A:ab e{ -'?-); (b) 2aV-4«.^^0\ l); (c)f7ra3-4,3^ a- ^ ^2; (7) Find the mean distance of points uniformly distributed along the perimeter of an ellipse from a focus. Aiis. One half of the major axis. UlAP. XVll.J TIlEUliV OF FUNUTIUNS. 283 CTTAPTER XVII. INTItOn'uCTIOX TO THE THE(MIY OK FUNCTIONS. 207. A function having but a single value for any given value, real or imaginary, of the variable is called a simjle-valued function. Rational Algebraic Functions, Exponential Func- tions, the direct Trigonometric Functions, and the Kllii)tic Functions are single-valued. A function which has in general two or more values for any given value of the variable is called a mnltiple-vahied function. Irrational Algebraic Functions, Logarithmic Functions, tiie inverse or anti-Trigonometric Functions, and the Elliptic In- tegrals, are multiple-valued. 208. In Chapter II. we have explained the customary graph- ical method of representing an imaginary by the position of a point in a plane, the rectangular coordinates of the point being the real term and the real coeflicient of the pure imaginary term of the imaginary in question. In the ordinary treatment of tlie Tlioory of Functions this metiiod of representation is of the greatest service, and enables us to bring the study of functions of imaginary variables within the province of Pure Geometry, and to give it great deliniteuess and precision. For the sake of brevity we shall in future use the symbol / for V — 1 and cis(^ for cos<^ -f- V— 1 sin<^, so that we shall write our ty|)ical imaginary as x + yi or as rcis<^, instead of using the longer forms .;• -f »/ V — 1 , and r (cos -(- V— 1 sln*^). We shall also use the name complex f/uautit;/ for an iniaginarv of the typical form when it is necessary to distinguish it from fi pure imaginary. 284 INTEGRAL CALCULUS. [Art. 209. 209. A complex, variable z = x-\-yi is said to vary co//////»- OMsly when it varies in such a manner that the path traced by the point (.T,y) representing it is a continuous line. Thus if z changes from the value a to the value y8, so that the point representing it traces any of the four lines in the figure, z varies continuously. It will be seen that a variable can pass from the first to the second of two given values, real or imaginary, by any one of an infinite number of different paths without discontinuity if the variable in question is not restricted to real values ; while a real variable can change continuously from one given value to another in but one way, since the point representing it is confined in its motion to the axis of reals. 210. A single-valued function w of a complex variable z is called a continuous function if the point representing it traces a continuous path whenever the point representing z traces a continuous path. A multiple-valued function of z is continuous if each of the n points representing values corresponding to a value of z traces a continuous path whenever z traces a continuous path. These n paths are in general distinct, but two or more of them will intersect whenever z passes through a value for which two or more of the n values of w, usually distinct, happen to coincide. Such a value of z is sometimes called a CiiAP. XVII] THEORY OF FUNX'TIONS. 285 critical value, and the consideration of critical valiu'.s plays an important part in the Theory of Functions. In studying a nniltiple-valued function we may confine our attention to any one of its n values, and except for the possible presence of critical points this value may be treated just as we treat a single-valued function. In representing graphically the changes produced in a func- tion tv by changing the variable z on which it depends, it is customary to avoid confusion by using separate sets of axes for w and z. 211. If we use the word function in its widest sense, w = it + vi will be a function of a complex variable z = x-\- v«, if u and v are any given functions of x and y. For example, 6?/, .^- + ?/^ x-yi, x^ -y' + 2xyi, X — y -^- xt Var^ + y- + 4 may all be regarded as functions of z. We have seen in Chapter II., Arts. .36-42, that with tliis definition of function the derivative with respect to z of a func- tion IV is in general indeterminate ; but that there are various functions of :, for instance, z"*, logz, e', s'mz, where the deriva- tive is not indeterminate. We are now ready to investigate more in detail the general question of the existence of a deter- minate derivative of a function of a complex variable. Let ?o = M + vi be a function of 2 ; u and v, which are real, being functions of x and y. Starting with the value Zo = x^)-\-y^,i of z and the correspond- ing value Wo = "o + ^'ui of w, let us change z by giving to x increment Ax without changing y. Let A^'< and A^v be the corresponding increments of « and V ; and 2i and Vi the new values of :; ami iv. 286 We have Tlieii INTEORAL CALCULUS. [Art. 211. ^0+ -•'•' v\, + Aj« + I'A^V. ^x Ax and the derivative of "• with respect to z under the given cir- cumstances is ^~=o[':;::]=--'^^^- [1] ■To If, however, starting with tlie same value z;„ of z, we change z by giving ^ the increment Ay without changing x, we have ;2i = A, + (.'/. + A.v) / = z„ + /A?/, and W, - Wo AyM lAyV ~ iA//"*" /A?/ Zj - Zo limit ~"'i-^^'"1_/) A2=0 z, — z,. * [2] and this is tiie derivative of w with respect to z when we cluinge y and do not change a;. Comparing [1] with [2], we see that if we start with a given value of z, and change z in the two different ways just con- sidered, the limits of the ratios of the corresponding changes in w to the changes in z need not be the same. Indeed, the two values for — given in [1] and [2] will not be the same unless to = 11+ 'vi is sucli a function of z = x + yi that J)^n = I\r Mild Il,„ = ~D^r. [3] CllAl'. XVII.] THKOllV OF FINCTKJNS. 2S7 We shall now show that if w is such :i function of z that tlio same if we equations [3] are satisiied, v-^^o — ^^''^ ^'*^ start with a given value z^y of z, no uiatttcr in what manner z may change ; that is, no matter in what direction the point representing z may be supposed to move ; or. in other words, no matter what may be the value of . . ,v r We have in general, since w is a function of the two variables X and y, \iv = (D, H + ilJ, r) A.I- + ( I)^ n + iD^ r) A// + t, where e is an infinitesimal of higher order than Ax- or A//. (I., Art. Ut8.) A2 = A.r + i\y. Hence \w_D^u.\x+ iD^ v.\y-\- iD, v . A.r +D^u.:^y-\- (. Az Ax + iAy A.'- A.r Ax Ax limit AZ: nit rA?<.'~| _ dw = ^l^]~ dz alue involving a'"^o " ' -'"'^ therefore dei)endent ui)on the direction in which z is made to move. If, however, [3] is satisfied, [4] reduces to ilz and the derivative of ?'.' is independent af ^^ nit fAvl ^<>La:J 288 INTEGRAL CALCULUS. [Art. 212. A function which satisfios equations [3], and which, there- fore, has a derivative whose value depends only upon the vahie of the independent variable, and not upon tlie direction in which the point representing the variable is supposed to move, is called by some writers a monoijenic finictiou, by others a. function ichlrh has a derivative. 212. Any function of n which can be formed by performing an analytic operation or series of operations upon « as a whole, without introducing x and i/ except as they occur in z, is a monogenic function of z. Yor if w =fz =f(x 4- yi) , where /2 can be formed by operating upon z as a whole, D^iv=f'z, and Dytv =if'z\ therefore iD^ic = Dyio, or iD^ (u + -^0 = D^ {n + tri) ; wlience D^n = D^v, and DyU = — D^v\ and [3], Art. 211, is satisfied. Consequently w is monogenic. This accouuts for the results of Arts. 38-42. If w is a multiple-valued function of z, there may be several diflFerent values of — , corresponding to the same value of z ; dz but if 'W is uionogenic, each of these values depends only upon z.- and not upon the way in which z is supposed to change. In future, unless something is said to the contrary, we shall give the name function only to monogenic functions. Thus we sludl not call such expressions as x — yi, or si? + y^ + 2xyiy functions of z. Conjugate Functions. 213. If n and v are functions of x and y, satisfying equations [3], \\{. 211, it is easy to prove that D^^ u + D^^u = Q and D^ v -f Z>> = 0- Chap. XVll.] THEOIIY OF FUNHrriONS. 289 For since D^ u = I)^ r :iik1 IJ^ c = — D^ii, we have D/ u = D^D^v ami D;h = - nj),v, D/ V = - I), I)^ n and /r- /• = 1>^ I)^ „ ; u and V are then sohitions of I>aphice's equation, I)/V+D;V=Q. [1] Any two functions <^ and ^ oi x and y, such that ff){x,y) + ill; {x,y) is a niouogenic function of x + yi, are called conjugate functions ; and, by what has just been proved, each of a pair of conjugate functions is always a solution of Laplace's Equation [1]. Thus ar — y-, 2xy; e'cosy, e^s\u>/\ ^ log (.x- + ^-), tan '•''; X are three pairs of conjugate fuuftions, since x- — y--{-2xyi = {x -{• yiy, e* cosy + ie' sin y = 6'+»"', -i log (x^ + y') + i tan~*^ = log {x + yi) , and consequently, by Art. 212, are all monogenic. Therefore each of the six functions at the beginning of this paragraph is a solution of Laplace's Equation [1]. It is clear that we can form pairs of conjugate functions at pleasure by merely forming functions of x + yi and breaking them up into their real parts, and their pure imaginary parts ; that is, throwing them into the typical form u -\- ri. If each of a pair of conjugate functions, <^ and tp, is written equal to a constant, the equations thus formed will represent a pair of curves which intersect at right angles. For let (.r, y) be a point of intersection of the curves = a, ip = b ; the slopes of the two curves at (.r, y) are respectivelv — — ^, — — ^ by I., Art. 202; and since D^4>=D^\p and iJ^ij/ = — D^, the second slope is minus the recii)rocal of the first, and the curves are perpendicular to each other at the point in question. Thus or — v/2 = a, 2xy= h, cut each other orthogonally ; as do 290 LNTECJKAL CALCULUS. [Art. 214. also ^\og{x' + y-)=a, tan ^•' = h\ or, what amounts to the same thing, ar + ir = a^ ^ = hi. It must be observed, how- o; ever, that ar + y- and - are not conjugate functions, and that X in general the converse of our proposition does not hold. It may be easily proved that if and ip are conjugate func- tions of X and y, and / and F are any second pair of conjugate functions of a; and y, the new pair of functions formed by re- placing X and y in <^ and ij/ by / and F respectively will be conjugate. Thus (e'cosjf/)- — (e'sin//)-, 2e'cos?/.e'sin2/, or, reducing, e^cos2y, e^s\n2y, are conjugate functions ; ^log [(o;^- fr + V2xyy-], tan-'^-^), or, reducing, log (x- + //-') . tan ' ( -^^X are conjugate. The properties of conjugate functions given in this article are of great importance in many branches of Matliematical Physics. Example. Show that if x' and y' are conjugate functions of x and y, X and y are conjugate functions of x' and y'. l^reacrvation of Anr/lcs. 214. If w is a single-valued' monogenic function of z, and the point rei)resenting z traces two arcs intersecting at a given angle, the corresponding arcs traced by the point representing tc will in general intersect at the same angle. CiiAr. XVII.] THEORY OF FUNCTIONS. For let Zq be the point of intersection of the cnrves in the 2 plane, and ?'•„ the corrosponding point in tlie it- plane. Let 2, be a point on the tirst curve, and z., a point on the second ; and let ^9<=:^^, Wi and lOo be the corresponding points in the ?o figure. Let ri, /'a, Si, and s.^ be the moduli of «, — Zq, Zo — Zq^ ju, — ?fo, and W2 — 10^ respectively, <^i, <^o, i/^i, and 1/^2 their arguments : then, since to is a monogenic function of z, we must have lim it r!!i^i^i= limit r^^viii^i limit r^ii^n= limit r^^^i^^i; \_i\ cis ^ij |_?-2 cis <^2 J whence, by Art. 23, limit —cis (t/'i — <^i) = limit - cis (i/^^ — <^2) » and since, when two imaginarit's are equal, tlicir moduli must be equal, and their arguments must be equal, unless the moduli are both zero or both inlinite, limit {\p., — i/r,) = limit (<^2 — ,) ; that is, the angle between the arcs in the in figure is equal to the angle between the corresponding arcs in the 2 figure ; unless [g..="' - m.-.: If 10 is a multiple-valued monogenic fimction of 2, and if Starting from any point 2o, the [joint which represents z traces 292 LNTEGUAL CALCULUS. [Art. 215, out two curves intersecting at an angle a, each of the n points representing the corresponding values of to will trace out a pair of curves intersecting at the angle a ; unless Zq is a point at which — is zero or infinite. dz If, then, w is any monogenic function of z, and the point representing z is made to trace out any figure however complex, the point rei)resenting w will trace out a figure in which all the angles occurring in the z figure are preserved unchanged, except those having their vertices at points representing values of z which make — zero or infinite. dz This principle leads to the following working rule for trans- forming any given figure into another, in which the angles are preserved unchanged. Substitute x' and y' for x and y in the equations of the curves which compose the given figure, x' and j' being any pair of conjugate functions (Art. 213) of x and y, and the new equations thus obtained will represent a set of curves forming a second figure in which all the angles of the given figure are preserved unchanged, except those having their vertices at points at which D^x' and D^y' are both zero, or at which one of them is infinite. For exniiiple, x — y = a, (1) x+y=b, (2) are a pair of perpendicular right lines. Replace x by xr — y' and y by 2 xy, and we get aJ_2a'j/-?/2 = a, (3) a^+2.T?/-y- = 6, (4) a pair of hyi)crl)<)las that cut orthogonally. 215. If ?o is a sinc/le-vcdued continuous function of z, it is clear th.it if Wq and v\ are the values corresponding to z„ and Z], CiLiP. XVII.] Til EOUV OF FUNCTIONS. 293 aud the point z moves from Zq to rj by two different paths, the corresponding paths traced by xv) will begin at ii\ and end at jc,, and consequently that if z describes any closed contour, w also will describe a closed contour. If ?c is a double-valued function of «, since to each value of z there will correspond two values of w, it is conceivable that if n.\ and Wx are the values of ?o corresponding to ^i, and z moves from 2o to 2j by two different paths, v:; may in one case move from Wq to t(7i, and in the other case from rv, to w^ . It can be proved, however, that if the two paths traced by z do not enclose a critical j^oint (Art. 210), and w is finite and continuous for the portion of the plane considered, tliis will not t ike place, and that the two paths starting from Wf^ will terminate at the same point tvi. We give a proof for the case where ;s is a single-valued function of iv. As z traces the first path, each of the two points repre- senting the two values of lo will trace a path, one starting at icq, and the other at tco, and unless the z path passes through a critical point, the two tv paths will not intersect, but will be entirely separate and distinct, and will lead, one from ivq to icx, the other from iVo to tVi'. If, now, the z path be gradually swung into a second position without changing its beginning or its end, since w is a continu- ous function, the two tv paths will be gradually swung into new positions ; but, provided that the z path in its changing does not at any time pass through a critical point, the two w paths will at no time intersect, and consequently it will be impossible for the w points to pass over from one path to tiie other, and there- fore the point which starts at «•„ must always come out at tf,, and not at ic/. It f()llows readily from this reasoning that if z describes a closed contour not embracing a critical point, each of the w points will describe a closed contour, and tiiese contours will not intersect. Of course, the proof given above holds for any nuiltiplc- valued function. In any portion of the plane, then, not containing critical 294 INTEGliAL CALCULUS. [Art. 210. points the separate values of a multiple-valued function may be separately considered, and may be regarded and treated as single-valued functions. 216. That in the case of a double-valued function two paths in the z plane, including between them a critical point, but having the same beginning and the same end, may lead to different values of the function, is easily shown by an example. Let w = Vz, and let z, starting with the value 1 , move to the value — 1 by the semi-circular path in the figure. That one of the corresponding values of v\ winch starts with + 1 will de- scribe the quadrant shown in the figure, and will reach the point 1 .cis-, If, however, z moves from +1 to — 1 by the semi-circular path in the second figure, the value of w which starts with + 1 will describe the quadrant shown in the second figure, and will reach the value l.cisf— ^J, or —i. These two paths described by z^ then, although beginning at the same point -{■ 1 and ending at tlie same point — 1, cause that value of the function which begins with + 1 to reach two different values ; and the two paths in question embrace the point 3 = 0, which is clearly a point at which the two values of jo, ordinarily different, coincide ; that is, a critical point. Chap. XVII.] THEOnV OF FUNCTIONS. OOi It is easily scon that if z, starting witli the vahie +1, de- scribes a complete circumference about the origin, the value of w which starts from the point + 1 will not describe a closed contour, but will move through a semi-circumference and end with the point I.cIstt or —1. Now, by Art. 215 any path described by z beginning with + 1 and ending with — 1 and passing above the origin, since it can be deformed into the semi-circumference of Fig. 1 without passing through a critical point, will cause the value of w beginning with + 1 to end with + i ; and any path described by z beginning with -{- 1 and end- ing with — 1 and passing below the origin, since it can be deformed into the semi-circumference of Fig. 2 without passing through a critical point, will cause the value of tv beginning with -f 1 to end with — i. Therefore any two paths described by z beginning with -fl and ending with —1 will, if they include the critical point z = between them, lead to diflferent values of IV, provided that the same value of lo is taken at the start. 217. If vj is a douldc-valuod function of z, and z describes a closed contour about a single critical point, this contour may be deformed into a circle about tiie critical point, and a line lead- ing from the starting point to the circumference of the circle, without afTecting the final value of w (Art. 215). Thus, in the figure, the two paths ABCDA, AB'C'D'B'A lead from the same initial to the same final vahu- of "• ; and this is true no matter how small the radius of the circle B'C'D'. 296 INTEGRAL CALCULUS. [Akt. 211 Let z<^ be the critical point, and let ir,, be the corresponding point in the w figure. As z moves from z^ towards Zq, the points representing the corresponding values of w will start at Wj and iL\' and move towards w^, tracing distinct paths. If, now, z describes a circumference about z^, and then returns along its original path to Zi, the first value of w will either make a complete revolution about ioq and return along the branch (1) to its initial value ^^'l, or it will describe about tu,, a path ending with the branch (2) of the w curve, and move along that branch to the value w,'. In the first case, and in that case only, the value of w describes a closed contour when z describes a closed contour, and is practically a single-valued function. If 2(1 is a point at which — is neither zero nor infinite dz {y. Art. 214), when z describes about z„ a circle of infinitesimal radius, to will make about ?<•„ a complete revolution ; for since if two radii are drawn from Zq, the curves corresponding to them will form at ?''o an angle equal to the angle between the radii, when a radius drawn to tlie moving point which is describing tlu' circle about Zy revolves thiough an angle of 3GU°, the cor- Chap. XVII.] THEORY OF FUNCTIONS. 297 iesi)uudiug liue joiniug h'o with the moving point representing lo will revolve throngh 360°, and we shall have what we have called Case I. If, then, we avoid the points at which — is zero or infinite, we shall avoid all critical points that can vitiate the results obtained by treating our double-valued or multiple-valued func- tions as we treat single-valued functions. A critical iwint of such a character that when z describes a closed contour about it the corresponding path traced by any one of the values of lo is not closed, we shall call a branch point. When a function is finite, continuous, and single-valued for all values of z lying in a given portion of the z plane, or when if multiple-valued it is finite and continuous, and has no branch points in the portion of the plane in question, it is said to be holomorphic in that portion of the plane. Definite Integrals. fz.dz was defined in Art. 80 in effect as follows : X > . dz = J'^'*^ [fz, {z, - zo) +fz, (z, - z,) + fz,{z, - z,) + - where z„ ^g? 2^3, . . . z„_i are values of z dividing the interval between Zq and Z into ?i parts, each of which is made to approach zero as its limit as n is indefinitely increased. is the line integral of fz (Art. IH.'i) taken along the straight line, joining 2,, and Z if Zg :""! ^ are repre- sented as in the Calculus of Iniaginaries. It has been proved that if /z is finite and continuous between Zq and Z, this integral dipeiids nuMely upon the initial and final values of z, and is equal to FZ-Fzo where Fz is the indefiuite integral \ fz.dz. 298 INTEGRAL CALCULUS. [Art. 218. If 2 is a complex variuble, and passes from Zoto Z along any given path, we shall still define the definite integral | fz.clz by [1] where now z,, Zg, z^, •••z„^i are points in the j^iven path. Two important results follow immediately from this defini- tion : 1 St. That (""fz .dz=~ C fz . dz, [2] if z ti'averses in each integral the same path connecting z,j and Z. 2d. That the modulus of I fz.dz is not greater than the line-integral of the modulus of fz taken along the given path joining z^t and Z. If we let fz — IV = u-\- vi, z = X -{- yi, u = (x, y) , and v = if; {x, y) , then I fz .dz— | ( u-\- vi) (dx + idy) =j {^, y) (J-^' + 'J "A {^, y) f'-*-' - J "A (^N y) '^y + 'J *^ (^'' ?/) ^y^ [3] each of the integrals in the last member ])eing the line-integral of a real function of real variables, taken along the given path connecting ^o and Z. If the given path is changed, each of the integrals in the last member of [3] will in general change, and the value of I fz . dz will change ; and, since z may pass from Zq to Z by an infinite number of different paths, we have no reason to expect that I /z.fZz will in general be determinate. We shall, however, i)rove that in a large and ini[)ortant class fz.dz is determinate, and de|)ends for its value upon Zn and Z, and not at all upon the nature of the path traversed bv z in <>;oin<>- from 2,, to Z. CiiAP. XVII.] TIIEOUY OV FUNCTIONS. 290 -I'J. li fz is holomorphic in :i -'~- [1] if z describes any closed contour lying wholly within that portion of the plane. From [;5], Art. 2iri. wo have j \fz . dz = iiv. (h = j ndx + / j vdx — Cvdij + / Cmh/, [2] the integral in each case being tlie line-integral aiounil t!ie closed contour iu question. Since w z= fz is holomorphic, u = ,dxd;/ ; the integral in the first nieniber of each equation l)eing taken around the contour, and that in tiie second member being a surface-integral taken over the surface bouuded by the contour. We have, then, from [2], C'fz.dz = f C(D^n + I)A')dxdy-\- i f f(D^r-I)^H)dxdy, [3] but D,u = DyV, and D^u = - D,c from [.}], Art. 211. Tlierefore, [3] reduces to j "fz . dz = 0. From this result we get easily the verv important fact that if fz.dz will have the same value for all paths leading from z„ to %, provided they lie wholly in the given part of the plane. For let z^/iZ and z^hZ be any two paths not intesecting between r„ and Z. Then z^aZbz^ is a closed con- ^ tour, and 300 INTEGRAL CALCULUS. [Akt. 220 I fz.dz (along ^o^'^^^^o) = r fz.dz (along z„aZ) -f C'fz.dz (along Zhz^) ■=()', but I "fz.dz {aXong Zhz^)) = — j fz.dz (along 2!„6Z) by Art. 218. Therefore, | /^.r/^ (along Zq^-Z^) = I /^. f/2; (along z^ft.^). If the paths ZqCiZ and z^ftZ inter- sect, a third path z^^^cZ may be drawn not intersecting either of them, and liy the proof just given Jfz . dz (along ZuaZ)= i fz.dz (along z^cZ) , I fz.dz (along 2oi-^) = j fz.dz (along z„f-^) ; therefore, X fz.dz (along Zo^-^) = ) fz-dz (along ZuftZ). 220. If fz, while in other respects holomorphic in a given portion of the plane, becomes infinite for a value T of z, then i fz.dz taken around a closed contour embracing T, while not zero, is, however, equal to the integral taken around any other closed path surrounding T. For let ABCD be any closed con- tour about T. With T as a centre, and a radius e, describe a circumfer- ence, taking e so small that the cir- cumference lies wholly within^4BCZ>. Join the two contours by a line ^Ll'. Then ABCDAA'D'C'B'A'A is a closed path within which /^ is holo- morphic. Chap. XVII.] THEORY OF FUNCTIONS. 301 Therefore, ([fz . (h (along ABCDAA'D'C'B'A'A) = 0, or Cfz . dz (along ABC DA) + Cfz . dz (along .Li') +J}> . dz (along A'D'C'B'A') + Cfz . dz (along A'A) = ; but Cfz . dz (along ^.4') =- Cfz. dz along {A' A) , id Cfz . dz (along A'D'C'B'A') = - Cfz. dz (along A'B'C'D'A') ence Cfz . dz (along ABCDA) = f/^ . (/^ (along A'B'C'D'A'). and Hence 221. That the integral of a function of z around a closed contour enihraciug a point at which the function is infinite is not necessarily zero is easily shown by an example. fz = , t being a given constant, is single-valued, con- z — t tinuous, and finite throughout the whole of the plane except at the point t, at which becomes infinite, without, however, z—t ceasing to be single-valued. /dz around a circle whose centre is f, and z-t whose radius is any arbitrarily chosen value c. If z is on the circumference of this circle z — t = (. (cos <^ -i- / sin <^) = o'" by [.^], An. .-.l. z=-t + ce''"' and . 302 INTEGRAL CALCULUS. [Akt. 221. Hence f-^ (around ahc) = T" ^^^^ = 2 wi. J: From what lias been proved in Art. 220, it follows that dz around any closed contour embracing t must also be z — t equal to 2iTi. ■ dz, when Fz is z — t supposed to be holoraorphic in the portion of the plane con- sidered, and where the integral is to be taken around any closed contour embracing the point z = t. — — is holomorphic except at the point z = t^ where it becomes infinite. The required integral is, then, equal to the integral around a circumference described from the point t as a centre, with any given radius e, that is, by the reasoning just used in the case of ( — — , to J z — t Jo £6*' */» and in this expression e may be taken at pleasure. If now c is made infinitesimal te** is infinitesimal, and since Fz is continu- ous F{t -\- ee*') is equal to i^^ + t; where ?/ is some infinitesimal, and F{t + ce*') d<^ is equal to Ft . c^ -{- -q . d<^. Now, by I. Art. IGl, f"" {Ft . dcf> + 7;(?<^) = C'^'fi . d = i i '"Ft . d = 2 iriFt ; and we get the important result that | dz, taken around any contour including the point z = t, is ecpial to 'Im.Ft. From Ihis we have 1 /* Fz Ft = J- J^.dzi 2 TTtJ Z — t Cll.vr. XVII] THEORY OF FUNCTIONS. 303 and we see that a holomorphic function is determined every- ichere inside a dosed contour if its value is given at every point of the contour. If in the formula Ft = — C^ dz [11 2 TriJ z-t *- -• we change t to t -\- A^ we get A«=J- r>..ri/— ! L.A = _L f Fz.d..At_ •iTTiJ \z — t-\t Z — tj '2iriJ {z — t){z-t-\t) whence limi A<= [ A« J 27r J A< = [ (^ _ <) (2 _ « _ AOJ r,, 1 C Fz . dz , ,-, or Ft = I , ; I 2] and in like manner we get 27rij {z-ty •- ■' and in general F^''H= ^ ( , , [4] each of the integrals in these formulas being taken around a closed contonr lying wholly in that portion of the plane in which Fz is holomorphic, and enclosing the point z = t. Til. The integral of a holomorphic function along any given path is finite and determinate, for, by [3], Art. 218, it is equal to the sum of four line integrals, each of which is finite and determinate (Art. 16G). If a series iVq + «'i + "'2 + "") ^^'here m'o, ?/'i, ti\ •- are holo- morphic functions of z, is uniformly convergent for all values of z in a certain portion of the plane, the integral of the series along any given path lying in that porti(m of the plane /,< the series formed of the integrals of the terms of the given series along the path in rpiestion, and the neu< series is convergent. 304 INTEGRAL CALCULUS. [Aut. 223. For, let S = W(i -\- Wi -{- W2 -\ \- "'„ + "'„ + 1 -| — = Wo + ^^1 + «'2 H h ">. + ^'«) where E„ = iv„ + i-\- w„ + 2+-, and where by hypothesis 71 may be taken so great that the modulus of R^ is less than c for all values of z in the portion of the plane in question, e being a positive quantity taken in advance and as small as we please. CSclz = Cwo dz + Cwi dz H 1- fw^,, dz + C R„ dz for any given value of n. By the proposition at the beginning of this article. | Sdz along the given path is finite and determinate, as are also I w'o dz, I n\ dz, etc. The modulus of | R^dz is not greater than the line-integral along tlie given path of the modulus of R^ (v. Art. 218). If, now, n is taken sufficiently great, each value of the modulus of i?„ will be less than e; consequently each element of the cylindrical surface representing the line-integral of the modulus of i2„ will be less than e (v. Art. 166), and | R,^dz will be less in absolute value than c multiplied by the length of the path along wliich the integral is taken. Therefore, C Sdz = Cwo dz + Civi dz + Ciiu dz-\--', and, since the first member is finite and determinate, the second member is a convergent series. Taylor^ s and 3farlil=1^4--.. [2] -a (Z-a)- {Z-af {Z - a)* and the second member of [2] is a convergent series. Multiply [2] bv ^, and the series will still bo convergent for each value of z which Ave have to consider ; we get 1 FZ •2 Tri Z - t 27r/[_Z— a + {f FZ ' (Z-a)- ^ (Z-a)' [••'] 306 INTE(;UAL CALCl'LUS. [Aht. 223. Integrate now botli nu'inbers of [H] uroiiud the circumfer- ence, and we have J. 27ri iJ Z-t •'■^i[J Z-a ^ 'J (Z-h)' + <'-">'/(^'"^+-]^ w and, since each of the functions to be integrated is holoniorphic on the contour around which tlie integral is taken, and tlie second member of [3] is convergent, each integral will be Unite and determinate, and the second member of [4] will be con- vei-gent. Substituting in [4] the values obtained in Art. 221, [1], [2], [3], and [4], we have Ft = Fa + {t- a) Fa + i!-Z^F'a + itr_l^F"'a + .-. If the point z = a is at the origin, a = and [5] becomes Ft=Fo + tFo + il F'n + ^^ F"o + ..., [C] which is Maclauriu's Theorem. That [5] is merely a new form of Taylor's Theorem is easily Been if we let t — a = h, whence t = a -\-h. and [a] becomes F{a + h) =Fa + h F'a + ^f^F"a+^F"'a + .... [7] [fl] can, of course, be written Fz = Fo + zF'o + |1 F"o +~i'""o + -, [8] and [/i] as Fz = Fi + (z - a) F'a + ^~ ~ 'p^F"a + i^^p^V"'a + • • • ; Chap. XVII.] THEORY OF FUNCTIONS. 3()T aud we get the veiT important result that if a function of z /.•! holomorphic ivithin a circle n-hose centre is at the oriyin it may he developed by Maclaurin's Theorem^ and the development will Jiold, that is, the series will be convergent, for all vabtes of z lying icithin the circle. If a function of z is holomorphic within a circle described from z =a as a centre it can be developed by Taylor's Theorem into a series arranged according to powers of z — a, aud the development will hold for all values of z lying within the circle. The question of the convergeucy of either Taylor's or Maclaurin's Series for the case when z lies on the circum- ference of the circle needs special investigation, and will not be considered here. If the function which we wish to develop is single- valued, in drawing our circle of convei'gence we need avoid only those points at which the function becomes infinite ; but if it is multiple-valued we must avoid also those at which its derivative is zero or infinite (v. Art. 217). 224. "We are now able to investigate from a new point of view the question of the convergence of the series obtained by Taylor's and Maclaurin's Theorems in I. Chap. IX. Let us begin with the Binomial Theorem, (a) (a-f//)" = a" + 7(a» '/t -fn ^" ~ ^ ^ r-'/r + •-, [1] or, following the notation of [!»], Art. 220, z'^ = a"-\-mr-\z-a)-\- ''^'\-^\ ,"-\z-ar+.... [2] If n is a positive integer, 2" is holomorphic throughout the whole plane, and [2] holtls lor all values of z and a, and [1] for all values of a and h. If n is a negative integer, 2" is single-valueil, and it i.s liuito and continuous except for 2 = 0, where 2" becomes infinite. [2] is. then, convergent for all values of z lying within a circle described with a as a centre and passing through the origin ; 308 INTEGRAL CAL(^ULUS. [Art. 224. tbat is, for all values of z, such that mod {z — a) < mod a ; and consequently [1] holds if viodh <.moda. If 71 is a fraction, z" is multiple- valued, and our circle of convergence must avoid the points at which — becomes zero dz or infinite ; but as the origin is the only point of this character, the circle of convergence is the same as in the case last con- sidered, and [1] holds for all cases where modh'' = e' (cosy + is\nij) ([4J, Art. 31) is single-valued and continuous, and becomes infinite only when x=(x. Maclauriu's development for e' holds, then, for all finite values of z. (c) logz = log (r cis^) = logr + i (Art. 33) is finite and continuous throughout the whole plane. It is, however, multiple-valued, but its derivative - becomes infinite z only when z = 0, and does not become zero for any finite value of z. logz, then, can be developed into a convergent series, arranged according to powers of z — a, for all values of z within a circle having the centre a and passing through the origin ; that is, for all cases where mod (z — a)< mod a. If z — a=h, we get log(a + /0 = loga-h^-^, + ;^,--^^-h..., [3] [3] holding for all cases where modh <.moda. If a = 1 and h = z, we get log(l +z) = ^-- -[---- + •••, [4] wliich liolds for all values of z where mod z g ( 1 -f-'''), given in I. Art. 136, Ex. 2, holds if mod2;<7r. (3) Obtain the following developim-nts, mikI fin.l f-.r what real values of x they hold good : INTEGRAL CALCULUS. [Art. 224. #. , , , / ., , ., . , , X 1 x^ , 1.3 X^ ^f^ e'.coso; -^"(l+y;+»(3«-2)^-,+ = 1 + 2x^ Ax* 3! 4! ' ^ _31^ \ 6! •••/ \/) tannic (j/) (1 + 2x + 3.-^2)72 = 1 _ X- + 2.1'"' /, x~ , 4 a;* 31 a/* V 2! 4! , .r- , 2;tr"' , 9x^ 3 5 U() log l-.'r; , , , , .^"^ , ar' ' 3 5 // . , 2 , , 2' , , 2M7 , , 2«.31 , , 2^691 „ I ^) tiinx=x-\ x^-] x-'-l .»•' H x/-\ .t" ... \/^^ 3! 5! 7! 9! ^ U! (A-) X . etn X = \ — '- 2.T« 2a-^« 3 4.") iilf) 4725 93555 (0 log tanf +a; ) = lo«r tan +2.''+ x^-i x'-\ x' ••• \4 J ^ 1 3! 5! 7! / N ,i„x 1 , ^ , •*■" 3.<;* 8.*^ 3a-'' , 5()a-' , ^ ^ 1 ! 2 ! 4 ! 5 ! 6 ! 7 ! ^ 2!3!4! 5! 77 a.-^ 6! (o) (ver.in-.r =2f.r + i^ + ^-^ a^ 1^^ a^ N ^ ^ ^ '^ V 3.2 3.5 3 3.5.7 'i J 2 . 2a- , X- 2. 2—2 .>: -|- ;r 4 4 4 •J- 4- 4- ^^^ G-5.r+.x-=' 2-.T 3 -a; ^^2 ^^J V2'' S^J riiAi'. XVII.] THEORY OF FUNCTIONS. 31 1 Aiisicers. {(() —a 0, - ^ < X < ^ if 7i < ; (o) -oc) -V2<.i-/ = x — ii'ii\x, (0) dx-^ dx ' ^ x{\-xyj^- -2 ;/ = (), (7) B/z - crDJ^z = U, (9) are differential equations. Tlie order of a differential equation is the same as that of the derivative of highest order which appears in the equation. Equations (1), (2), (3), and (4) are of the first order ; (6), (7), (8), and (9) of the second order; and (o) of the fourth order. The degree of a differential equation is the same as the power to which the derivative of highest order in the equation is raised, that derivative being supposed to enter into the equation in a rational form. p:quations (1), (2), (3), (5), (6), (7), (S), and (9) are all of the first degree ; (4) is of the third degree. A differential equation is linear when it would be of the first degree if the dependent variable and all its derivatives were regarded as unknown quantities. Equations (2), (5), (G), (7), (8), and (9) are linear. The equation not containing differentials or derivatives, and expressing the most general relation between the primitive vari- ables consistent with the given differential equation, is called its general sohdion or complete prlmitioe. A general solution will always contain arbitrary constants or arbitrary functions. The differential equation is formed from the complete primi- tive by direct differentiation, or by differentiation and the subsequent elimination of constants or functions Itetween the primitive and tiie derived (-(luations. If it has been formed by ditferentiation onh/ without sub- sequent elimination or reduction, the differential ecpiation is said to be exact. 314 INTEGRAL CALCULUS. [Art. 227. A suigular solution of a differential equation is a relation between the primitive variables which satisfies the differential equation by means of the values whicli it gives to tlie deriva- tives, but which cannot be obtained from the complete primitive by giving particular values to the arbitrary constants. 227. We shall illustrate the use of the key by solving equa- tions (1), (2), (3), (4), (5), (6), (7), (8), and (9) of Art. 226 by its aid. (1) (l+x).v-f(l-y).x-^=0, or {\+x)ydx+{\-y)xdy=Q. clx Beginning at the beginning of the key, we see that we have a single equation, and hence look under I., p. 326 ; it involves ordinary derivatives : we are then directed to II., p. 326 ; it contains two variables : we go to III., p. 326 ; it is of the first order, IV., p. 326, and of the first degree, V., p. 326. It is reducible to the form --!— dx -i ■- dy = 0, X y which comes under Xdx + Ydy = 0. Hence we turn to (1), p. 330, and there find the specific direc- tions for its solution. Integrating each term separately, we get log x-\-x->r log ?/ — y = c, or log (xy) -{- x — y = Cy the required primitive equation. (2) x^^^-ay = x-\-l. Beginning again at the beginning of the key, we are directed through I., II., HI., IV., to V., p. 326. Looking under V., we see that it will come under either the third or tlie fourth head. Let us try the fourth; wo are referred to (4), p. 3.S0, for specific directions. Obeying instructions, the work is as follows : dx CUAr. XVIII.] DIFFERENTIAL EQUATIONS. KEY. .^15 xdii — ayiLr =0, d^i _ oiix _ , y ^^' log?/ — a log.v = c, log^'^ = c; x" a;" y =C.t-, (1) rfv /< . 1 , ..(10 dx d.i Substitute in the given equation, aCx" + x''+'^^- aCx" = .r -f- 1 , dx- a;»+i— -(.r+l) = 0, da; dC--^'+/f/a; = 0, (a — \)x" ' o.r ' Substitute this vaUie for C in (1), and we get \a a— I J the n'(iuired primitive. (3) x^ - V + A- V.r=-* - V- = 0. (/or ■ Beginning at the l)egiiuiing of the key, we are directed through I., II., III., IV., to v., pjigo 32G. Looking under V., we find tlmt our equation does not come under any of the special forms tlu-re given. We are consequently driven to obtaining a sohition in the form of a series, and for specific instructions we are referred to (13), page 332. Obeying these, our process is the following : 316 INTEGRAL CALCULUS. [Art. 22?. # _.V ,/72 -2 dyo _ .Vn _ W 2 _ -. 2 da; a: dxo .t„ -7-;^ — — y V.X — »/, -7—7 — — Z/o V .Tf, — T/o , cLc^ X dxo 3*0 da;' a; da^o a'o and the general value of ?/ is y = yo + {x- X,) (^1 _ V.V3^^) - ^^^^'(z/o + |/.V^^^) (a^ - a;n)'' /3 ?/o /-^ -A (a;-a;o)* / 4 /^ ,\ (x-a'n)^ /5 j/o /-;; , + -^T-[^'^ + ;^ ^^o-yo-J + ^T"(^ir - Vxv - yo This result can be very greatly simplified by breaking up the series ; we have -^ ^\ 2! 4! G! ( x-a-„) / (X - a;,,) '^ , (a; - a;,,) * (.f - x^^ ^•'""^T^l 2! +'"~4! (rr~ -^ V 3! 5! 7! / CiiAP. XVIII.] DIFFERENTIAL EQUATIONS. KEY. 31 - ViV ^•(^")(^^-'"'-'^^*''5T^' 7! ft V^ COS (x — a^o) — si'i {'C — a",i) m' '] = a;M^ cos (.r — a^o) — Jl — ^ • sin (x- — a-o) . |_a*o ^ ajy J ^ is entirely arbitrary ; call it sin a, then Xo y = .r[siua cos(a; — x^) — cosa sin(.r — .^\|)] = xsin[a— (a; — aV)], y = X sin (o — a;) , whore c is any constant. (4) ^'=4-^- Beginning at the beginning of the key, we are directed throngh I., II., III., IV., to VI., page 327. Looking under VI. we see that the equation is of the first degree in x ; we are referred to (17), page 334, for our specific instructions. Obeying these, we first replace — by p ; the equation becomes " dx ' p'^ = y* {y + •''"p) • Differentiate relatiyely to ?/. and we get ^j/h' = 4^ (y + a:^0 + ^!/' + -^V' v* dy dy Eliminate a;, dy y \ I'Jdy p (fy y 318 LNTEG UAL CALCULUS. [Akt. 227. Striking out the factor 2y.' + y"', we have 1^-^ = 0, P dy y a differential equation of the first order and degree in which the variables :iic separated, and which therefore can be solved by (1), page 314. Its solution is log\/> — log//- = C, P or — = c. y- Kliniinating p between this and the given equation, and re- ducing, we have c?/ (.r — c'-) = 1 , as our required solution. (5) 'l|_2^ + 2f?-2f' + y=l. (1) ax* clx^ dx- dx Beginning at the beginning of the key, we are directed through I., II., III., VII., to (22) (a),' page 335, for our specific directions. We see at once that y=l is a particular solution. Obeying directions, we have now to solve Let y = e""", and we have m* - 2 ?«•■' -f 2 m"^ -2m-\-\=0, as our auxiliary algebraic equation in m. Its roots are 1, 1, V^^, -V-H". The solution of (2) is then l( = (A + Bx) c' 4- C cos x-\- D sin .r, and of (1) is 2/ = ( J + Bx) c' + C cos ..• -f- D sin x + 1. Chap. XVIU.] DIFFEKENTIAL EgUATiuNS. KKV. 3U* (6) siu-.f , •„ + siiLi; COS X ,' — if = x — sin x. (1) (Lr ax Beginning at the beginning of the key, we are directed through I., II., III., VII., to (24), page 337, for onr specific instructions. Dividing through by siu-.r, the equation becomes cPy , ^ (/y , , ,n\ -r^ + ctn x-^ — CSC- X .y = x esc- x — esc x. (2) dxr clx y = etna; is found by inspection to be a solution of dry ^ (hi , „ --4 + ctn x~ — csc-.r . w = ; tZjf dx (2) can then be solved by (24) (a). Substitute y = 2;ctn.i- in (2), and it becomes ctu x — + (ctn- a; — 2 csc^x) — = x csc-.r — esc a;, rfar dx or — ■ — (tau.i; + secx'csca;) — = a; sec x esc a; — seer. (3) dir dx Referring to (25), page 339, and obeying instructions, we dz let z' = — , and (3) becomes dx dz' (tan.r + sec a; esc a*) z' = x seer cscr — seer, dx a linear differential equation of the first order in z'. whose solu- tion by (4), page 330. is z' = A tanr seer — r sec^r + tan .r secf (log tan" — log sin.r) ; but z' = — , whence integratiiiii^, we have dx z =B-{- Asecx — xtiinx — (1 -f seer) log (1 + cos.r), and y = A esc r + J5 ctn x — x — (esc .»• + ctn x ) 1< )g ( 1 -|- cos x) . 320 INTEGRAL CALCULUS. [Akt. 227. (7) .^i-x)-pl-2y = 0. d.i" Beginning at the beginning of the key, we are directed through I., II., III., VII., to (24), page 337, for our specilic instructions. Let us try the method of (24) (e), page 339. Assume y = '2,a„x"\ and substitute in the given equation; we have 2 [m (m — 1 ) a,„ af ^ — 2m {m — 1 ) a^cc"* + ?/i (m — 1 ) a„, 0?"*+^ — 2 a,„ a;"*] = 0. Writing the coefficient of ic"" in this sum equal to zero, we have m (m + 1) a„.+, - 2[m(m-l) +l]a„ + (m - l)(m - 2) o,„_i = 0, and we wish to choose the simplest set of vahies that will satisfy this relation. Substituting m = 0, ?h = — 1, m = — 2, etc., in this relation, we find a_i^c'o. a_2=a_i, tf_3 = (x_2, •••. Hence if we take Oo= ^i it follows that a_ 1 = a_2 = rt_:5 • • • = 0, and no negative powers of x will occur in our particular solution. Substituting now ?)i = 1, ?/i = 2, m = 3, etc., we have tti = ttg = «3 = O4 = • • • . Taking aj = 1, we get as our required particular solution of the given equation y =x + ar + .r' + ic* + •••. This can be written in finite form, since we know that 1 +.r + .x-2 + ar''... = — ? — 1—x Hence y = ^ l-x is a pailicular solution. CiiAP. XVTII.J DIFFERENTIAL EQUATIONS. KEY. 321 Turning now to (24) (a), page 337, we fiml y = r-^^ + C-' ( 1 + X + — -— log.x* ]. I —X \ i — X Beginning at the beginning of the key, we are directed tln-ough I., II., III., VII., to (24), page 337, for our specific instructions. Let us trv again the method (24) (e), page 339. Assume 2/ = Sa^x"", and substitute in the given equation, 2[??i {m — l)a„.af -- + a^.-c'" — 2 a„.x™ ^j = o. The terms containing af are (??i + 2) {m + 1 ) a^+'iX'^ + a,^x'^ — 'ia^.^^^ ; writing the sum of the coefficients equal to zero, we have m(m + 3)a„,^2 + «». = 0- (1) Letting m — and m = — 3, we get Oo = «'^ud a 3 = 0; and all terms of y involving even negative powers of x disappear, as do all terms involving odd negative powers, except the — 1st. In general «„-.o= (2) m {ill + 3) From this we get a2_ 1_ "' ~ 2^5 ~ 3 ! 1 •J! 11 Hence y = -^--^ + -— - -^~- 3 3 ! o 5 ! / / ! i) if we take a., a.. 2.4. .0.7 «2 2.4.5.6.7 .9 flj 2.4.5.0.7 .8.1) .11 7 -^- X* + - is a particular solution of the given equation. This can be tiirown into liuile form witlioiit nuicli hibor. J2 INTEGIl \L CALCULUS. We have xy _X^ 3 ' X' 3!5 +4 I 7 7!9 d{xy) dx = x2_ x' ~3! + 5! x^ x-'o 7!"^9! [Akt. 227. 9! 11 / x" , X? x'x^\ = .rsina; ; whence xy = sin. x- —a; cos re, and y = -(sina; — iccosa;). X Bj' going back to (2), and using odd vahies of m, we get another solution of our given equation, namely, — i_L^__£iL •^-"' x^ ^~ X 2 2!44!G 61S' which can be reduced to y = - (cos X + X sui X) . re Hence our complete solution is y = -[A{cosx-\- ajsin.r) 4--S(sin.r — rccosa;)], X y=AS2lll=^ + sin (x-cA if we let -4 = tauc. (9) DJ'z-a-DJ'z = 0. Uoginning at the beginning of the key, we are directed through 1. and IX. to (45), p. 347, for our specilic iustruc- tions. Chap. XVIII.] DIFFERENTIAL i:(HATl(tN8. KKV. 323 Obeying these, our work is iis follows : cly- — a-d.ir = 0, chj - adx = 0, (1) dy + adx = 0, (2) dj)dy-a-dqdx = 0. (3) Combining (1) ami (3), we get dpdy — adqdy = 0, or dp — ad(j = 0. (4) (1) gives y — ax = a. (4) gives p — oq = ^. (2) and (3) give us, in the same way, y-\-ax = ai, p + aq=(3i; and our two first integrals are p-aq=.t\{y-ax), (5) P + f(q=f,{y + ax), (6) /", and f, denoting arbitrary functions. Determining p and 7, from (o) and (G), P = ^ 1/2 {y + ox) +/, {y - ax) ] , g = -L [/, {y + ax) -/, {y - ax)] ; '2 a dz =itlMy^-(fx) +j\{y-a.,^-] ^'•'- +./^^ [./;.(.'/+"-*^')-/i(.y-«^)]^3/ /, {y + ax) ((/.y + adx) —J\ (// - ax) {dy - adx) ~ ' 2a Hence, z = F{y + ax) + F, (// - ax) , where F and F, denote arbitrary functions obtained by integral ing/i and/,, which are arbitrary. 324 INTKGUAL CALCULUS. [Akt. 228. 228. When a tliffeiential equation does not come under any of the forms giveu in the key, a change of dependent or inde- pendent variable, or of both, will often reduce it to one of the standard forms. No general rule can be laid down for such a substitution. It will, however, often suffice to introduce a new letter for the sum, or the difference, or the product, or the quotient of the variables, or for a power of one or of both. Sometimes an ingenious trigonometric substitution is effective, or a change from rectangular to polar coordinates ; that is, the introduction of rcos cf> for x and rsin^ for y. The following examples of such substitutions are instructive. (A.) Ch((vge of dei^endent variable. (1 ) (x -f- 1/)-'-^ — ((-, reduces to — /h — dx — U, dx a- + z- if we introduce z = x + //. (2) — = sin(c/) - 0) , reduces to — d = 0, c/<^ 1 — sin w if w=tf>-6. (;')) {x — y-)dx + 2 xifdii = 0, reduces to {x — z) dx + xdz =0, \i z= ?/-. (4) x^ly -y + x V.ir' - y' = 0, reduces to ^^^ + dx = 0, dx • Vl - z- ifz = y. (•^^) d-tf dx- '2 dy + - -f-n-ii = x dx 0, reduces dx- -n'z = 0, if z = Xl/. (B.) ClKwge of indej)e) dent variable. (1) (1- -.Tg + ,= 0, reduces to eos-^^ + sin^c -^>^ = 0, if X = sin 8. CiiAP. XVIII. j diffp:rential equations, key. 32c (2) ^^'>' + tau x^ + COS-.K . ^ = 0, reduces to '-^ + y = 0, cIt- dx ilz- if 2 = sin X. (C.) CIi(iur/e of both variables. to (1) (\- '^.rn = ^ (.r - y-" - (<■-') , reduce. ^^ dx-J ax V — z— — —(z — V — (('-') =0, if z = .r- and v = y- dz- dz^ ' dii ^ (2) (y - x) ( 1 + .1") ' ^ = ( 1 + //-) - , reduces to sin (<^ — ^) rf<^ = (/^, if .>-=taiii9 and // = tau/-\-c) dx + {a'x + h'y-\-c')'hj = (.3) 330 Linear. General form, -^ -\- XiV = X-,, where (Ix Xi and X, are functions of a; alone * ... (1) 330 Of the form ^ + Xjy = X^y'\ where Xj and X dx are functions of .r alone * (5> 331 Mdx + Xdy an exact differential. Test, D^M = D,N (Gj 331 Mx + Ny = (7) 331 Mx-Ny = Q (.s) 331 Of the form F, (.r?/) ydx + F. (xy)xdy = . (H) 331 D,M-D,N ^ ^ fuuetion of x alone . . . (10) 331 N ^'^~ Dy^^ ^ a function of y alone . . . (11)332 M D^M- D,N ^ ^ function of (xy) (12)332 Ny — Jfx A solution in the form of a series can always be obtained . (13) 332 VI. Not of first degree. Can be solved as an algebraic equation in p, where » stands for — (14) 333 dx Involves only one of the variables and p, where » stands for — (lo) 333 dx Of the first degree in x and y ; that is, of the form xfip + yf22)=f3P-, where p stands for ^ (IG) 333 dx ^ J Of the first degree in x or // (17) 334 Homogeneous relatively to x and // . . . . (18) 334 • Of cour«t', A', ami A', may be ciiiiHtantM. 328 INTE(;iiAL CALCULUS. Page Of the form F{(fi,if/) = 0, where ^ and ip are functious of X, ?/, and — , such that 4> = a ^ dx \pz=h, will lead, on differentiation, to the same differential equation of the second order (ID) 334 A singular solution will answer (20) 334 VII. Not of first order. Linear, with constant coefficients ; second member zero* (21) 335 Linear, with constant coeflicients ; sec^z -}- TD^^z=V, where R, S, T, and V may be functions of x, y, z, IJ^z, and D^z . . . (!.'>) .347 X. Containing three variables (II) 340 Containing more than three varial>les . . . (12) .34(1 XL Containing tinve variables (43) 'MG Containing more than three variables . . . (44) 347 • Sec note, p. 310. 330 INTEGIIAL CALCULUS. (1) Of or reducHjle to the form Xdx + Ydy = 0, where X is a function of x alone and I'is a function of y alone. Integrate each term separately, and write the sum of their integrals equal to an arbitrary constant. (2) M and X homogeneous functions of x and y of the same degree. Introduce in place of y the new Aariable v defined by the equation y =■ vx, and the equation thus obtained can be solved by (1). * Or, nuiltii)ly the equation through by , and its ^ ^ ^ ^ -^ Mx-\- Xy first member will become an exact differential, and the solution may be obtained by (6). (3) Of the form {ax + by + c) dx + (a'x + b'y + c') dy = 0. If ab'— a'b = 0, the equation may be thrown into the form (ax + by + c) dx -\ — (ax + by + c)dy = 0. If now z = ax-{- by be introduced in place of either x or y, the resulting equation can be solved by (1). If ab' — a'b does not equal zero, the equation can be made homogeneous 1))' assuming x = x'— a, ?/ = y'—fi, and determining a and (S so tliat the constant terms in the new values of 3/ and X shall disappear, and it can then be solved by (2). (4) Linear. General form --\- Xiy= X.2^ where X, and dx Xj are functions of x alone. Solve on the supposition that X2 = by (1); and from this solution obtain a value for y, involving of course an arl)itrary constant C. Substitute tliis value of y in the given equation, regarding C as a variable, and there will result a differential equation, invohnng C and x, whose solution by (1) will express C as a function of x. Sub- stitute this value for C in the expression already obtained for y, and the result will be the required solution. KEY. 381 (5) Of the form -• + XiV = X. y", wIktc X, aud X, arc dx functions of x alone. Divide through by y", and then intnKhiee 2=?/'"" in place of ?/, and the equation will become linear and may be solved by (4). (6) Mdx + my an exact differential. Test D^ M = D, N. Find I Mdx, regarding y as constant, and add an arl>i- trary function of y. Determine this function of y by the fact that the differential of the result just mentioned, taken on the supposition that x is constant, must equal Ndy. Write equal to an arbitrary constant the | Mdx above mentioned plus the function of y just determined. (7) Mx + Ny = 0. Divide the first term of Mdx + Ndy = by 3/x, and the second by its equal —Ny^ and integrate by (1). (8) Mx - Ny = 0. Divide the first term of 3Tdx + Ndy = by Mx, and the second by its equal Ny, and integrate by ( 1 ) . (9) Of the form /, {xy) ydx +f^ (xy) xdy = 0. Multiply through bv , and the first mem])t'i ■' Mx — Ny will become an exact differential. The solution may then be found by (0). (10) ^J^ ^^, a function of x alone. -^ rng M-n.y ^^ Multiply the equation through by e^ s ' ', and the first member will become an exact differential. Pho solution may then be found by (G). 332 INTEGRAL CALCULUS. (11) — ^ -—-^ — , a function of y alone. Multiply the equation through by e^ m '"^^ and the first member will become an exact differential. The solution may then be found by (6). (12) — * i — , a function of (xy). Multiply the equation through by e*^ Ny-Mx ' " where V = xy, and the first member will become an exact differ- ential. The solution may thus be found by (6). (13) A solution oiMdx + Ndy=^Q in the form of a series can always be o])tained. Throw the given equation into the form -M = — - — clx J X then differentiate, and in the result replace — by — — , thus obtaining a value of ^^ in terms of x Jy dxr and y ; b}- successive differentiations and substitutions d^ 11 d^ ?/ get values of — -i, —4, etc., in terms of a; and y. dxr dx* If 2/0 is the value of y corresiwnding to any chosen value .To of X, y can now be developed by Taylor's Theorem. We have y =fx =/(•>•„ -}- x — X(,) Xo + - = /»•„ + {X - x,)f% + (•^-^•o)> r,„ + i^^ofj^nr dxo 2 ! dx„- 3 ! d.V where ^, ^, ^, etc., dxQ dxo^ rfa'o" are obtained by replacing x and y by Xq and y^ m the values of , ,2 -, dy^ (Py^ c^^ gj^^^ dx' dx^' djr'' described al)ove. KEY. 333 In the general case ?/u •» entirely arbitrary, and if the given equation is at all complicated, the solution is apt to be too complicated to be of much service. If, however, in a special problem the value of y corresponding to some value of X is given, and these values are taken as //o and iCu, the solution will generally be useful. (14) Can be solved as an algebraic equation in j), wlu-re p stands for -^. dx Solve as an algebraic equation in j), and, after trans- posing all the terms to the first member, express the fust member as the product of factors of the first order and degree. Write each of these factors separately equal to zero, and find its solution in the form V— c = by (V.) . Write the product of the first members of these solutions equal to zero, using the same arbitrary constant in each. (15) Involves only one of the variables and p, where 2> stands for^. dx By algebraic solution express the variable as an expli- cit function of j>, and then differentiate through relatively to tlie other variable, regarding }) as a new variable and remembering that — ^ = -. There will result a differen- dy p tial equation of the first order and degree between the second variable and p which can be solved by (1). Eliminate j) between this solution and the given equation, and the resulting equation will be the required solution. (16) Of the form xf^p -f yf.,p =f^p, where p stands for ' j!.. Differentiate the equation relatively to one of the vari- ables, regarding p as a new variable, and, with the aid of the given equation, eliminate the other original variable. Tiiere will result a linear dilf'erential equation of th*- first 334 INTEGRAL CALCULUS. order between p aud the remaining variable, which may be simplified by striking out any factor not containing -^ or -^, and can be solved bv (4). Eliminate p between this dy solution and the given equation, aud the result will be the required solution. (17) Of the first degree in x or y. The equation can sometimes be solved by the method of (If)), differentiating relativel}' to the variable which does not enter to the first degree. (18) Homogeneous relatively to x and y. Let y = vx, and solve algebraically relatively to p or -y, p standing for -^. The result will be of the form p — fv, dx or v = Fp. If J. dy ~ d(vx) ~ dv , ^ dx dx dx an equation that can be solved by (1). If 'o = Fp, y- = Fx>, y = xFp, X an equation that can be solved by (16). (lU) Of the form F((f}, if/) = 0, where (f> aud ij/ are functions of X, y, and ^, such that = a and U/ = b will lead, on dx differentiation, to the same differential equations of the second order. Eliminate — between = a and if/ = b, where a aud b dx are arbitrar}' constants subject to the relation that F{a, b) = 0, and the result will be the required solution. (20) Singular solution will answer. Let -^=2i, and express 2^ as an explicit function of x d and ?/. Take — , regarding x as constant, and see ^ dy "" ° kf:y. 385 whether it can be made infiuite by writing eijual to zero any expression involving y. If so, and if the eciuatiou thus formed will satisfy the given differential equation, it IS a singular solution. <,') Or take —^J-L, regarding y as constant, and see whether it can be made infinite by writing equal to zero any ex- pression involving x. If so, and if the equation thus formed is consistent with the given equation, it is a singular solution. (21) Linear, with constant coetficieuts. Second member zero. Assume y = e'"^ \ m being constant, substitute in the given equation, and then divide through by e"". There will result an algebraic equation in m. Solve this equa- tion, and the complete value of y will consist of a series of terms characterized as follows : For every distinct real value of m there will be a term Ce"" ; for each pair of imaginary values, a-f6V — 1, a — 6 V — 1, a term Ae"' cos hx + Be"' sin hx ; each of the coefficients A, B, and C being an arbitrary constant, if the root or pair of roots occurs but once ; and an algebraic jjolynomial in x of the (r— l)st degree with arbitrary constant coetlicieuts, if the root or i)air of roots occurs /• times. (22) Linear, with constant coellleieuts. Second member not zero. (a) If a particular solution of the given equation can be obtained by inspection, this value plus the value of »/ detained by (21) on the hypothesis that the second mem- ber is zero, will be the complete value of tlie dependent variable. 336 INTEGRAL CALCULUS. (b) If the second member of the given equation can be got rid of by differentiation, or by differeutiatiou and elimination between the given and the derived equations, solve the new differential equation thus obtained, by (21), and determine the su[)erfluous arbitrary constants so that the given equation shall be satisfied. In determining these superfluous constants, it will generally save labor to solve the original equation on the hyi)othesis that its second member is zero, and then to strike out from the preceding solution the terms Avhich are duplicates of the ones in the second solution before proceeding to differentiate, as from the nature of the case they would drop out in the course of the work. (c) If the given equation is of the second order, solve on the hypothesis that the second member is zero, by (21), obtain from this solution a simple particular solution by letting one of the arbitrary constants equal zero and the other equal unity, and \ety = v be this last solution ; then substitute vz for y in the given equation ; there will result a differential equation of the second order between x and z in wliich the dependent variable z will be wanting, and which can be completely solved by (25). Substitute the value of z thus obtained in y = vz and there will result the required solution of the given equa- tion. (d) Solve, on the hypothesis tliat the second member is zero, and obtain the comiilete value of y by (21). Denoting the order of the given equation by n, form the n— 1 successive derivatives -ii, — -1 ... ±. Then dx f?.x~' dx"~^ differentiate y and each of the values just obtained, re- garding the arl)itrary constants as new variables, and sul)stitute the resulting values in the given equation ; and by its aid, and that of the v — 1 equations of condition formed liv writinir each of the derivatives of the second set, KEY. 837 except the nth, equal to the derivative of the same order in the first set, determine the arlntrary coelileients and sub- stitute their values in tlie original expression for >/. (23) Of the form (a + bx) "'^ + A {a -{- bx) '-' 'j"^' + • • • + /.// = X, where X is a function of r alone. Assume a-{-bx = e', and change the independent vari- able in the given equation so as to introduce t in place of X. The solution can then be obtained by (22). (24) Linear; of second order; cocllicieiits not constants. General form ' ■" '' + I' '['' + Q>/ = li. (Li- (Ix (a) If a particular solution y = v of the equation can be found by inspection or other means, substitute y=cz in the given equation, which will then reduce to the form v'^^ + (2'l^ + Pi\'^ = R, dar \ (Ix J .(Ix and can be solved by (2.")). Substitute the value of z thus found in y t'z, and the result will Ik- the general solution of the given ecjuation. (b) The substitution of y = rz in tiie given equation, wliere v is given by the auxiliary ditTi-rential ecpiation 2''- + I>r = 0, dx INTEGRAL CALCULITS. and can be found by (1), and should be used in the simplest possible form, will lead to a differential equation in z of the form which is often simpler tluui the original equation. (c) The introduction of z in place of the i:idc|H'n(lont variable x, z being a solution of the auxiliary differential equation da^ dx the simpler the better, will reduce the given equation to the form which is often simi)ler than the original e(piation. {d) If the first member of the given equation regarded as an operation performed on y Can be resolved into the product of two operations, tlie equation can always be solved. The conditions of such a resolution are the following; let the given equation be dx- dx where w, ?', w, and R are functions of x ; this can be resolved into >+")(''£+'')-"=-'^' where/), 7, r, and .s are functions of .r, if /(/;- , \ , , da j)r = ?<, 'jr -{-J) I 1- .s ) = V, and (/s -j- }> — • = ■<<;; \(lx J dx and the values of p, 7, i\ and s can usually be ()l)tainod KEY. bv inspection. Wo have first to solve p \-i-ee 0, -- of the degree — 1, •••. ax " (/.r Assume x = e^, y = e^z, and by clianging the varialUes introduce 6 and z i^to the equation in the place of x and >/. Divide through by e* and there will result an equation involving onlv z, -, — -, •••, whose order iii:iv lie de- (16 de- pressed by (2.3). (31) Homogeneous on the supposition that x is of the degree 1, 7/ of the degree ?(, '' of the degree ?i — 1, - •'' of the clx rf.ir degree /i — 2, •••. Assume x = e^, y=ze"^z, and by changing the variables introduce and z into tlie equation in the place of x and //. The resulting equation may be freed from 6 by division and treated by (25). (32) Homogeneous relativelv to ?/, ~, — -,•••• dx dxr Assume i/ = e', and substitute in the given equation. Divide through by e* and treat by (25). (33) Containing the first power only of tiie derivative of the highest order. The equation may be exnrt. Call its first member — . H n is tlie and _ -^ by ^ . Multiplv th*- it-rni ' (/.»;"-' ^ ' dx" ' dx containing ' ^' h\ were the onlv ''■'' ' d" 'v variable, calling the result T', ; tiien replacing /i by .^~, 342 INTEGRAL CALCULUS. (Ijr fiud the complete derivative ', and form the expression — -', representing it hv ^. U - — ? contains the dx dx " dx dx first power only of the highest derivative of ?/, it may itself be an exact derivative, and is to be treated pre- dV ciselv as the first member of the given equation — has dx been. Continue tliis process until a remaintiier '^ of dx the first order occurs. Write this equal to zero, and see if the equation thus formed is exad^ see (G). If so, solve it by (6), throwing its solution into the form F„_i = C A complete first integral of the given equation will be Vi-\- Ui+ ■■■ + V„i= C. The occurrence at any step of the process of a remainder — ^, containing a higher power than the first of its highest derivative of y, or the failure of the resulting equation of the first order above described to be exacts shows that the first member of the given equation was not an exact derivative, and that this method will not apply. (34) Of the form '^^x'^+ y\'^;''^ =0, where X is a (/ar dx L"-'J function of x alone and Y a function of y alone. jNIultiplv '[: and may be solved by (33). through by [ - •' | and the eciuation will become exact, (3;")) Singular integral will answer. Call ^9), and ^ 7, and find ^, regarding « and o da;""' dx" dp dq as the onl}' variables, and see whether ~J- can be made d}) infinite by writing i-qtial to zero any factor containing p. 343 If so, oliiniuato 7 hctwoon this equation and the pjivon ♦.'(luation, and if the result is u sohitiun it will be a singular iute solved by (;>.S), Eliminate the new variables representing the various derivatives from the ecjuations of the solution, and tlie equations obtained will be the complete primitive sys- t.'m recjuired. (40) All the partial derivntives taken with respect to one of the independent variables. 346 INTEGRAL CALCULUS. Integrate by (II.) as if that one were the only indepen- dent variable, replacing each arbitrary constant by an arbitrary function of the other independent variables. ^^41) Of the first order and linear, containing three variables. General form, l*I)^z + QD^z = R. Form the auxiliary system of ordinary differential equa- W/j* (111 (Tz tions — = -^ = — , and mtegrate by (38), Express their primitives in the form u = a, v = b, a and b being arbi- trary constants ; and ?< =/y, where /is an arbitrary func- tion, will be the required solution. (42) Of the first order and linear, containing more than three variables. General form, P^Dx^z -f P^Dx^z -f- = i?, where a;,, a-g, , cc„, are the independent and z the depen- dent variables. Form tlie auxiliary system of ordinary differential equa- tions ^L^J:^^}^ = ^' = 1?5, and integrate them by (38) . Ex])ress their primitives in the form v^ = a, u. = &, Vg = c, and Vj =■ f{^V).2^v^-, ,v„), where /is an arbitrary func- tion, will be the required solution. (43) Of the first order and not linear, containing three varia- bles, F(a;,?/,2;,/),7) =0, where p = D^z^ q = D^z. Express 7 in terms of cc, y, z and p from the given equa- tion, and substitute its value thus obtained in the auxil- lary system of ordinary differential equations _^ = ^V = ; — = — . Deduce by integration from the.se equations, by (30), a value of p involving an arbi- trary constant, and substitute it with the corresponding value of 7 in the equation dz = pdx -f qdy. Integrate this result l>y (3G), if possible; and if a single primitive equation be obtained, it will be a complete primitive of the given equation. KEY. :)[! A sinn:iilar solution ma}' be ohtainod by finding tiie partial derivatives D^z and D^z from the given equation, writing them separately equal to zero, and eliminaliiig j) and q between them and the given equation. (44) Of the first order and not linear, containing more tlian three variables. F{Xi,x., , .t„, z.p^.p.,, ,p,^) = (J, wla're Pi = Dx,z, 2h=J^x,z. Form the linear partial ditTcrential equation 5j[(Z).r(F + PiD.F)Dp - Dp.FiDj:.^',„i?ii iPn)i ^'^f^ where 2,- means the sum of all the terms of the given form that can be obtained by giving i successively the values 1, 2, 3, , n. Form, b3'(42), its auxiliary system of ordinary differen- tial equations, and from them get, by (38), n — 1 mte- gi-als, i = (fi, o = a.,, ,4>„_i = a„_i. B}' these equations and the given equation express j^i, j^a^ ■> /'« in terms of the original variables, and substitute their values in tlie equation ch = pi (/.r, + p., dr., -f- +p„ fU',.. Integrate tins b}' (37), and the result will be the required complete primi- tive. (45) Of the second order and containing the derivatives of the second order only in the first degree. General form, RD/z + SDJ)^z + TD^z = T", where li, ^', T, and I'may be functions of .x', ?/, 2, Z>,2, and D^z. Call I),z p and D^z q. Form fu'st the ecjuation /.',/,/-■ _ Si] .all, + T(h- = 0, [11 and resolve it, sup[)osing the first menilier not a complete; square, into two efiuations of the fijrm (hj — )ii , il.r = 0, d>/ — m.,(lc = 0. [2] From the first of these, and from th<' equation Rdpdii -I- Tdqdj- - \ 'dud'f = 0, [3] 348 INTEGRAL CALCULUS. combined if noodful with the equation dz = pdx + qdy, seek to obtain two integrals ?f, = a, Vi = fS. Proceed- ing in the same wa}' with tlie second equation of [2], seek two otlier integrals u., = a„ v., = ^i ; then the first in- tegrals of the proposed ecjuation will be « V INTEGRAL CALCULUS. -. {U)l xdx + ydy)+ ^Ay^lV^ = 0. Ans. ^+^' + tan^^^ = c, ' \ ' ■ aj- + i/- 2 a; yl»s. {'2y — x^ — c) [log (x- + y — \) + x — c\ =0. '^"^' (^ ~ I ~ 7 V ^~~^ (^^»^ - 1 - c W 0- (19) ri-?/^-^;y^^Y-2^^+<=o. ory yaxy x* ox* xr ^y/.5, 2/ + log — -^-^ — ^" !/ — log -^ — c 1 = 0. \ y J\ y (20) y = ic^^ -(- 1:^ _ T-l^ ) . Ans. y = ex + c — y^) y dX dX yj^^^y . • _U 1 V f\ ^ p /. i Singular solutiou, y =^ '—■ 4 1) 2/ = 2/ t~ +2^*/- /^ , ^ns. ?/- = 2ca; + c-. rZxv dx M./l* '^^> [>-(:!!)>= <^-'-«=);;: L^^O <->-^-t+<;;! jifff'f (24) ar^v' +^-.V •'+^■'' = '// Urr-^-\/x- Ans. 7/2 - c.r- --I- -^^-^ = 0. 1+c ^1/iS. y- = 2ca; + c^. ^Hs. p- + cry + a^.T = 0. , ^f-Ci 351 (25) / (26) X Ans. {b + i/Y = 4 ax, /(a) -f 6 = 0. ,/- - x>/'^''~\- Ans. -^ = 6, r(a) + b = 0. " dx y — a dx xlns. y = ((\, + c, X + Ca^r + Cj ar') e'. ^bis; y^- + {A + Bx) cosx + (C+ />a;) sino,-. (30) ^_2^ + 4v = f?'cosx. dar' d.i- yl/(,s. y = Ae ^ + el (J5— "^ Jcosa;+^C + ]-^Jsina- . ^^rT" d*y ^ d^y d?y _ , d-t;-" dor^ du? Ans. y= {A + Bx) e' + {C + Dx)-\-\)>x- + 2,.r^ + |- + •20 v^<^ 4f^ + 4^ = .-^. "^* Ans. y = {A + Bx)e-^+\{-2.i- + Ax-{-Z). (34) ^ + 4.v = a;sin-x ^Ins. ?/ = ^1 cos a- + 2J siu .c -f- - sin x. Ans. y=f A- -^^cos2x + (b- p \ sin ^ J: + ^ ^-^/ + 2y = xlogx. d.r Ans. y = ^Ix cos (lof? .r) + 7>V sin (log^^ . . 7 ) ''" '' + - '-^ - »-' V = 0. Ans. v = - (Ae'" + Be ") . .7.'- X dx ' ' X ZA — '' ' ■ Ans. y = A cos (sin x) + B sin (sinx). (39) {\-x^y'ly, + y=.Q. dx- , . . Ans. y = VT^^^ (A + B 1( )g ^-^' (40) {\+x')'^-2x'^ + 2y = i). ''•*-■" ^^" .'L^s. y = Bx + .1(1- x') . ,.,- d-y X dy , 1 , (41 ) , — ^ ^ H y = x — \. dx^ X — I dx x — I . i , , r> / 1 . ^\ xhis. y = Ae' + Bx — {\ + x^) (42) ^pl, - -^a- (1 + .^) 'hL^-2{\+ .r) y = x\ dx- dx „ Ans y = A.ve'' + Bx • 2 (43) siu-.c^^ -2y = 0. Ans. y =A ctn.c -\- B (l - a;ctna;). (I-O ^(+-vi^ ,/ = .'f- + loga-\ dx- X- log x \a; J Aw^. y = A log a: + e^ log.f + J3( loga; | — x V -^ logo; /4rN f , ( „ ,^n(\\ ^''^d^-x-^dx^{:'--'-^)^'= Ans. y = e"^lA + -^,+ "^ JT" ' 2 (2a - 1) (46)!''?--^''i'+f,.'+4,y,=o. dx- x dx \ x-j . , . , 7) • \ ^ ^ x\ IIS. y = a- ( A cos (/x 4 B sin ax) . KEY. 353 (47) g-2to^+6Vy=0. Arts. y = e'^{Acosx-\/b + Bs\ux\/b). (48)f:f-4./'" + C4:r--:i).. = ,.". Alts, y = e' {Ac^ -{- B*" —1). (49) (l_ar')g-4.T|J-'^-(l4-ar^)y=.r. ] * Aiis. >/ = {x + .1 COS.C 4- B sin.'-). 1 — X- (50) 'i!^--L ^^•'/4-^±j/^ii^v = o. rfx^' ^y^ dx Axr ' An.^. y = e'''4Ax- + ?-\ (51 ) ^-^ + 2 » ctn //.r-'^ + (m* - ^i^) y = o. (/x- dx t , . , T> • x ^ln.s. y = (^1 cos ?im 4-ijeos dor X dx X* ' x x (54) d'y 3a; + l (/y f 6(x+]) T_ ^ (?x-' .1- - 1 f/.r • L ( .r - 1 ) ( 3 a- + 5 ) J Ans. 2/ = [.l4-i>'log((x- iy(3x + 5))]V(x- l)f»(3a; + 5). (65) (l-^,-^'-.-^-c-,v=0. Ans. y= Ae"'" ' + Be''"" ''. (5G) (H-f,a-)^ + ax^-n2»/ = 0. dxr dx Ans. y = A{^l+ax'-^-xy/ay^-hBiy/l+ax' + x \fa) ^. (57) (X - 1 ) (X - 2 ) 'f •' - ( 2 .r - ;^ ) '1^ + 2 .V = 0. d.i" dx Ans. 2/ = c (X - 2)'^ -h c' (X - 2) [(X - 2) log (x - 2) - 1 ]. 354 INTKcaiAL CALCULUS. (58) (3 - X) ''"'? - (9 - 4a;) ^ + (6 - 3a:) // = 0. Alls, y = re^ + c'e^i x -] ar ar). (59) (a''-x-)'^'y-Hx'^-\2j/ = 0. '^•^' '^"^ . c ^ ,(a- + 3x2) Ans. y = - „ + c- '• Ans. y = — [A (sin nx — nx cos nx) + B (cos nx + yia; sin nx) ] . ar (61) ^+li^^ = 0. ^Hs. y = cloga; + c'. dx- a; dx (62) Ar'i? + .^'j/^f?! + 4 .,/SY+ 2:.yf? = 0. \ dx J dx- Y<-y dx Ans. cr -\- cxy = c' X. (03) (x^ + 2,"^yf| + 2,/*)'+ 34^ + i, = 0. V dxjdx^ \dxj dx t^. , « * ■ . i ^ ^ ^ -^ iMiid a first integral. Ans. x--=^ + 2/^1 -^ ) + av/ = c, dx \dfa;/ (64).r-!^+:.f| + (2.r,-l)';'' + ../- = 0. rind a first integral. Ans. X? — -^ — x-^ + xy^ = c. d.v dx 1 ( G5 ) -^ + ^y- + — ~ = 0. Ans. {x-a){>/-b)(z-c)= c. 1/ • X —JM y — b z —c _ 6) -{y + z)dx -\-dy + dz = 0. Ans. e' (y + z) =c. (67)^ + 4x4-^ = 0, ^ + 3y-x=0. dt 4 dt .^ ^ , J Ana. x = ce -i -•' y = (ct+Ci)e 2. ^^-^^^^ . V KEY. 866 (68) ^ + m'x=0, ^-vrx = 0. (G9) D,z=-y—. x-\-z h(S. X = A sin m( + B cos mt, x + y = Of + D. Ans. e~* (.c + V + 2) = (fii/. ( 70) xzD^z + i/zD z = Qcy. Ans. z- = xy -f- (-] \yJ (71 ) D,z.D^z=l. Ans. z = a.r + -^ + b. a (72) j-D/z-\-->x>/D,D^z + yWJ^z = 0. Ans. z = xz + ij/z. (74) D,z.D,D^z-D^z.DiZ = 0. Ans. X = y + {l/z. APPENDIX. Chap. V.J INTEGRATION. 66 CHAPTER V. INTEGRATION. 74. We are now al>le to exteiKl iiuiU'rially our list of formulas for direct integration (Art. 55), one of which may l>e obtained from each of the derivative formulas in our last chapter. The following set contains the most important of these : — Z>, log X = - gives /, - = log X. ' '=' X ^ ^^x "" Z),a' = a'logc " /^a'log(( = «'. D^e' = e' " f^e^ = e\ D^sin.f = cosx " /^cos.t = sin-c. i>,cosa.' = — sinx " fA~ sin.r) = cosa*. Z),logsina; = ctnx " y'^ctn.f = log.sin.r. Z>,logcosa; = — tanx " f{~ tan.f) = logcosx. DMn-'x = ^ — - - /; ' ■=— '• /;jan-'.i-=— ! — " /;_l_=t:in-'j-. 1 + ,'»•=' 1 -f- .'- l)^ \ers- ' .r = ' • f, — = vers" ' j-. V(2x--x-') • VC-^-^--^) The second, fifth, and seventh in the second group can Ixj written in the more convenient forms, 66 DIFFERENTIAL CALCULUS. fABT. 76. /.«' a- loga ' f^s\nx= — cosx; . y^tanx = — log cos X. 75. When tlic expression to be integrated does not come under any of the forms in the preceding list, it can often be prepared for integration by a suitable change of variable, the new variable, of course, being a function of the old. This method is called integration by substitution, and is based upon a formula easily deduced from D^ (Fy) =D^Fy.D^y\ which gives immediately Fy=f^{D^Fy,D^y). Let u=D,Fy, then Fy=f,u, and we have f,n=f{uD^y)', or, interchanging x and y, Ln=f,{uD,x). [1] For example, required f{a + bxY. by [1] ; Let z = a + bx, and then f{a + bxy ^/^z" =f{z" . D,x) but H-l --h hence Ma + bxy = lrz'^=]. ?^. 6 6 n-l- 1 Chap. V.] INTEGRATION. 67 Su])stituting for z its value, we have ,;(„+,,.,.).=i(i±M:::. P^XAMPLK. Fiiiil /; — ?— . Ana. i log(a + /-/J-) • a ■\-hx h 7(!. If /;<• represents a funetioii that ean be integrated, /((i + ^j*) can always be integrated ; for, if 2 = a -f- ^^1 then D,x = - b and fj{a + hx) = fjz= f, fzD, x=-l\fz. b EXAMPLES. Find (1) f^smax. ^-Ih.s. cos ax. a (2) y; cos ax. 4 1 • Ans. -sin«jr, a (3) /.tanax. (4) f.cinax. % 77. Ri'fiuiredf, 1 -^y /.- sJia^- -ej -• Lot a tlien xz=az, D,x = o, 68 DIFFERENTIAL CALCULUS. [AuT. 78. /. ' V(l-^-) a Examples. Find 1 1 a; 78. Required/^ ^ Let z = x + VC-^' + «^) ; z^ — 2 z;.r + .j" = or + a^ 2zx = z- — a^, ^2 _ „i 2z ' ^(a:^ -f , v + vD^ v , which gives uv =J\ uD^v +f^vD^xi or f^uD^v = nv—f^vD^u. [1] This method is called integrating by parts. (a) For example, required /^log.r. log a; can be regarded as the product of logo; by 1. CaU log. I' = u and 1 = B^ f, then I)^u = -, X v = x\ and we have y;iog.c =f 1 log.r =J\ uD^v = ny — /, t'-^i" = d;log.r —JV- = x\ogx — .r. X PlXAMPLK. Find /,a* log a;. Suggestion: Let loga;= » and .r= D^v. Ans. ^a-(logx-l) 80. Required f^9\v?x. Let u = sin.i- and />^/*=slnx, ihcn />^» = cosx', v= — coax, /,sin*x = — sin.rci)s.>- +./;cos'x ; 70 DIFFERENTIAL CALCULUS. [Art. 81. but cos^a; = 1 — sin^ic, so XcCOS^x=f^l—/^shr'x = x—f^6m'x and y^sin^a; = x — sinx'coso; — y^siu^a;. 2Xsin^a; = x — sin a; cos a;. Xsin^a; = J (a; — sin a; cos a;). Examples. (1) Find/^cos^a;. Ans. -(.'c + sin.Tco.sx). (2) /,sinxcosx. Atis. ^-. 81. Veiy often both methods described above are required in the same integration. (a) Required f,s\xr'^x. Let sin'"';c = y, * then a;=sin?/; DyX — cos?/, Xsin-'.r =./;?/ =,/;?/ cosy. Let u = y and D^ v = cosy ; then Z)^?A = 1, v = siny, and /yycosy=ysin7/— /,sin?/=?/siny+cosy=a:sin-'a;+ V(l— ar'). An}' inverse or anti-function can be integrated by this method if the direct function is integrable. {h) Thus, fJ-'x=j^y=f^yDJy = yfy-fJy where y=/-»x. Chap. V.J INTEGRATION. Tl Examples. (1) FindXcos~*a;. Ans. a;cos-*a; — ^(1 — x*). (2) /,tan-*x. Ahs. a-tair'.i; — l()ii(l 4- jr). ( o ) /, vers" ^x. Ans. (x — l) vers" ^x -{- ^{■2x —jr). 82. Sometimes an algebraic transformation, either alone or in combination with the preceding methods, is useful. (a) Required/. x-^-a^ ar— a- 2 a \x — a x + a J and, by Art. 75 (Ex.), ^C^. 1 + X _ 1 __j_ V(i-a-=^) V(i--^") V(i--'^) V(i--'-) can be readily obtained by substiditing 2/ = (1 — x*), and is — V(l — -i") ". hence /, J(jzf^) = ^i""'-^' - V(l - ^) • (c) Required /yj(^ii'-x'). 72 Dll^FKKENTLVL CALCULUS. [Art. 83. and f,^{a' -x^) = «y. .^ -J\ -~^ whence /, V («' - ^") = ^'' «i»~ ' - -/« , f .„ , by Art. 77 ; o. V(^' — ^'") but /.VCa'' - ar^) = x^{a' - or) +./; ^^ f ,, , 6y integration by imrts^ if we let ti = ^ (a^ — x-^) and D^v=l. Adding our two equations, we have 2y;V((r - oy") = x-^{a? - s?) + n^siu-i ^ ; and .-./xVC"^ - •*■') = -^Ma^-o? + a^sin-^^Y Examples. Find (1) /.V(.^-' + «'). (2) /,V(x-2-a2). 1 -d«.s. - [x-^J^x^ — a-) — «'■ log(x- + Vor — a^) J. Applications. 83, To find the area of a segment of a circle. Let the equation of the circle be x^ + ?r = "^» and lot the required segment be cut off by the double ordinates through (a;o,2/o) and (.<",?/) . Then the required area A=2f^y-i-C. Cii.vp. v.] INTEGKATION. 73 From the equation of the eiivle, y = V(*r -.'-"), hence A = 2/, V ("" - -t" ) + C] ami therefore, by Art. 82 (*•) , A = x-^ia- - .1") + tr sin-' '^ + C. As the area is measured from thc^ ordinate »/„ to tlie onlinatt' y, ^1 = when .r = Jo ; therefore = .r„^( .r =:-L =IL^ by Art. 73 ; D^y m s = —fy ^rn'+ y- ^:^\jJ V"^' + / + »'i'log(?/ + V//r+y-)] + ^i by Art. 82, Ex. 1. If the arc is measured from the vertex, s = when ?/ = ; 0=-i-(m-logm) + C, 2 ?u C= — -m\og7n, and ,^l pV('«' + /) „|„, y + V(m' + /) ]. 2 [_ 7«, ^ O ^j, J Example. Find the length of the arc of the curve .r^= 27,?/- inchidcd be- tween the origin and the point whose abscissa is 15. Ans. 19. P^^ ^^^^^^ y^^^ 'i^-e-X •^ .