310 
 
 P4 
 1914 
 
 UC-NRLF 
 
9-A SHORT TABLE OF INTEGRALS 
 
 COMPILED BY 
 
 B. O. PEIRCE 
 
 // 
 
 LATB HOLLIS PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY 
 IN HARVARD UNIVERSITY 
 
 ^^ 
 
 ABRIDGED EDITION 
 
 GINN AND COMPANY 
 
 BOSTON . NEW YORK • CHICAGO • LONDON 
 ATLANTA DALLAS • COLUMBUS • SAN FRANCISCO 
 
/D 
 
 > • • • 
 
 • • • • 
 
 • • • •• 
 
 • . • • • • 
 
 
 COPYRIGHT, 1914, BY 
 GINN AND COMPANY 
 
 ALL RIGHTS RESERVED 
 
 PRINTED IN THE UNITED STATES OF AMERICA 
 
 329.4 
 
 GINN AND COMPANY • PRO- 
 PRIETORS • BOSTON • U.S.A. 
 
m 
 
 FUNDAMENTAL EQUATIONS 
 
 1. I a 'f(x)dx = a I f(x)dx; I <f>('i/)dx= I ^^^ dy, where y'=di//dx. 
 
 2 . I (u-\-v)dx=j udx-{- j V dx, where u and v are any functions of ar. 
 
 3. I tcdv = uv — I vdu ; lu-T-dx = uv— iv-^- dx. 
 J J J dx J dx 
 
 4. / x'^dx = — ——7 f if m ^ — I'j j — = log x, or log (— x). 
 
 5. I e'^dx = e'^/a ; / b'^dx = — • 
 
 6. I sinxdx =— Gosx; lGOSxdx = smx. 
 I tan xdx = — log cos x ; i ctn xdx = log si 
 I sec^ajc^o; = tancc; / 
 
 sina;. 
 
 '.dx =■ tan X ; I csc^icc^o; = — ctn x. 
 
 i. I coshccc^o; = sinho"; / sinh ic c^a? = cosh cc. 
 I tanh xdx = log cosh cc ; / ctnh x = log sinh 
 
 ®* / "Tl — s = -tan-M-)j or ctn-M-)- 
 
 , J d + x^ a \a/ a \a/ 
 
 /dx 1 /x\ 1 (X -h ic 
 -^ 5 = - tanh-M - 1 j or 77- log 
 a^ — x^ a \a/ 2 a °a — x 
 
 /dx 1 ^ , - /x\ 1 - X — a 
 -i -9. = ctnh-^ - )> or ~— log • 
 x^ — a* a \a/ 2a ^ x-\-a 
 
 3 
 
 857200 
 
9. f_^==sin-(^), 
 
 ... J, VaV-^. . ; : V«/ 
 
 FUNDAMENTAL EQUATIONS 
 
 ©■ 
 
 or — COS" 
 
 '^^' "'=log(a^,+ V^^T^). 
 
 /c?x 1 ' Ja\ 
 =-cos-M-)- 
 
 r /^^ = _ 1 i.^/ ci + v^^±^ \ 
 
 J x^a'±x^ ^ \ X / 
 
 10. r_^ 
 
 a \ — a 
 
 — 2 
 
 or — ;p=:tanh" 
 
 1 f^ 
 
 —J or — pctnh- 
 
 ^Af 
 
 4-^a; 
 
 11. e^ = cos X + i sin cc ; e""^ = cos x — i sin a;. 
 
 12. sinh x = i(e' - e"^) ; cosh a; = |(e^ + e"^). 
 
 13. sin xi = i sinh a? ; cos xi = cosh x. 
 
 14. sin X = — i sinh xi ; cos x = cosh xi. 
 
 15. log u = log (c2^) — log c. 
 
 16. log cc = log (- x) + (2k 4- 1) TTi ; log.x = (2.3025851) • log^^x. 
 
 17. log (x + yi) = log r -\- <f>i, x — r cos <^, ?/ = r sin <^. 
 
 sin (x + yi) = sin x cosh y -^r i cos x sinh ?/ ; ^^^^ 
 
 cos (x + yv) = cos X cosh ?/ — * sin x sinh 2/. ^^ 
 
 18. sin-^x 4- cos-^x = — ; 
 
 tan~^x + ctn~^x 
 
 19. sinh-^x = log(x + Vx'-^H-l); cosh-^x = log(x + Vx^— l); 
 
 tanh~^x = \ log ; 
 
 J. — X 
 
 ctnh~-^x = -J- log ; 
 
 X — -L 
 
 sech~^x = log 
 
 1+ VT 
 
 csch'^^x = log 
 
 +vr+ 
 
 20. ctnh"-^x = tanh~-^x -\- \iri -\- kiri 
 
EATIONAL ALGEBRAIC FUNCTIONS 5 
 
 ^ I. RATIONAL ALGEBRAIC FUNCTIONS 
 
 A. Expressions involving (a -f- hx) 
 The substitution of y or « for x, where y — xz = a -{- bx, gives 
 
 22. { x{a + bxydx = - ly'^{y - a)dy. 
 
 23. J x^ (a 4- ^'a^)-c^a^ = -^ij^ (^ - ^T^^- 
 
 2^ r x-dx ^ 1 ny-aydy 
 J {a-^bxy b^' + ^J y"" 
 
 25 r ^ = i_r(i^^:^ 
 
 » ' J x"" (a + bx)"" a'^+^-^J «»* 
 
 'df« 
 
 Whence 
 
 dA. 27 f— ^^— =--^— . 
 
 28 C ^^ = 1 
 
 J {a + bxf 2b(a-\-bxy' 
 
 /xdx 1 
 ^IfT^ = ^[« + ^^ - alog(«^ + bx)l 
 
6 . EATIONAL ALGEBRAIC FUNCTIONS 
 
 J (a-h bxf b^l a-\-bx^ 2{a + bxf\' 
 
 /a?dx in 1 
 
 33. 
 
 /x^dx If a^ 1 
 
 ^^^^, = -^\a + bx-2a log(« + to) - ^^^J^ 
 
 / dx _1 
 x{a-{-bx)~'~ a^^ 
 
 g^ , -- 1 , a + Sa; 
 
 35 C ^^ — 1 _ i_ , «_+ 
 
 J x(a + bxf ^ a(a + bx)~ a"- ^^ x 
 
 36 C ^^ — i _L. ^ 1 ^ + ^^ 
 
 J x^{a-\-bx) ax a^ ^ x 
 
 B. Expressions involving (a + 5af) 
 ^Q r dx 1 c + ic r dx 1 a; — c 
 
 ot^ C dx 1 x^fab 
 
 J a-\-bx^ -slab a 
 
 At\ r ^^ 1 , a-\-x^ — ab 
 
 J a + &c2 2^-ab "" a-x-yJ-ab 
 
 1 , , , 05 V— ab 1 ^ , .x^—ah 
 
 or . tann~ ^ j or . ctnn~ ^ • 
 
 V— a6 « V— a5 « 
 
 * j {a + ^>ic2)2 ■" 2 a(a H- bx^) 2^ J ^ 
 
 
 ^ 
 
RATIONAL ALGEBRAIC FUNCTIONS 
 
 r dx 1 X 2m — 1 r dx 
 
 ^' J (S^ + bx'y^'^ ~2ma(a-h bx'Y 2ma J (a-{- bx^ 
 
 , r xdx _ 1 /^ ^^ r — 2T 
 
 J {a-^-bx'y^^ ~ 2J (a + bzY^^' Iz = x \. 
 
 AK r ___dx__^ _ _1_ , ^ 
 
 • J x{a + bx') ~ 2^ ^^ a 4- 
 J a-^-bx" ~b~bj a + bx^ 
 
 43. 
 
 44. 
 
 dx 
 
 -^-bay" 
 
 dx 
 
 * ^ ar^(a + ftar^) ax aj a 
 
 48 r ^'^^ ^ -^ I ^ r 
 
 49 r dx _ 1 r_^_dx b r dx 
 
 ^T ^cx C dx 1 , iC" 
 
 52. I = — loar • 
 
 J x(a + bx"") an ^ a -\-bx'' 
 
 r dx _ 1 r dx _ b_ r x'^dx 
 
 • J (a + ^aj«)'»+i ~" aJ (a H- te")"* a J (a + ^>ir")'"+i ' 
 
 r x'^dx _ 1 r g;"*"" a ^ cc^'-^tZa; 
 55 C ^^ - 1 r ^^ ^ r c^a; 
 
RATIONAL ALGEBKAIC FUNCTIONS 
 
 -i 
 
 a. 
 + 
 
 ■I 
 
 + 
 
 
 
 + 
 
 + 
 
 + 
 
 a. 
 
 IT 
 + 
 
 + 
 
 a. 
 
 IT 
 
 + 
 
 g 
 
 
 g 
 
 
 g 
 
 
 « , 
 
 , ^ , 
 
 , s , , 
 
 1 
 
 
 ^.^ 
 
 
 
 .--^ 
 
 
 i^ 
 
 
 1^ 
 
 -11 - 
 
 + 
 
 H 
 
 + 
 
 tH 
 
 + 
 
 
 v___ 
 
 ►C 
 
 
 
 
 00 
 
 O 
 
 I— ( 
 
 W 
 
 W 
 X 
 
 O 
 
 + 
 
 + 
 11 
 
 t 
 
 I 
 
 I" 
 
 I 
 
 I 
 
 Ch 
 
EATIONAL ALGEBRAIC FUNCTIONS 
 
 |Mb -_ Cxdx 1 1 ,. h Cdx 
 
 -- Cxdx bx -\- 2 a b Ti 
 
 «3j^= — ^ -J 
 
 r xdx _ 2a-^bx _ b(2n-l) Cdx 
 J Z«+i~ nqX"" nq J X"' 
 
 ^_ f x% X h . ^^ b^-2ac rdx 
 
 ^^ r^ 7 (b^-2ac)x-{-ab , 2a rdx 
 
 66. l—^dx = ^ ^ 1 I — • 
 
 J X^ cqX q J X 
 
 ^« C x'^dx _ g;"'-^ n — m-\-l b C x'^-'^dx 
 
 J JC» + i"~(2n-m-f-l)c^"~27i-m + l cj Z«+i 
 
 m — 1 a r x"*-?dx 
 
 2n-m-\-l'~cJ Z'»+i 
 
 ^o f (^£c 1 , ar^ b rdx 
 
 „ f ^a^ 5.x 1 ^/ b' c\ rdx 
 
 70 f— ^— - ?^ n-{-m — l b^ r dx 
 
 ^m-l^»+l 
 
 m-1 a / a;'»-2jc'»+i' 
 
10 RATIONAL ALGEBRAIC FUNCTIONS 
 
 D. Rational Fractions 
 Every proper fraction can be represented by the general form ; 
 
 f{x) ^ g^x-^ + g^x-^-{- g^x-^+ -"+9, 
 F(x) a:" + A;jCc«-i + k^x""-^ H \-k^ ' 
 
 If a, b, c, etc. are the roots of the equation F(x) = 0, so that 
 
 F(x) = (x — ay (x — by (x — cy---, 
 
 fix) A, A^ A^ A 
 
 then ^774 = 7 ^ + 7 '-r—:-^ . \. . + -"+ ' 
 
 F(x) (x-ay (x-ay-'^ (x-ay-^ x-a 
 
 4- ^^ + ^ + ^ + -.-+-^ 
 
 ^ (x- by ^ (x- by-'' (x- by-^ ^ ^ x-b 
 c, c„ c, c\ 
 
 + 7 -Tr-^7 V^ + 7 V^ + --- + 
 
 (x — cy (x — cy~'' (x — cy~'^ x — c 
 + , 
 
 where the numerators of the separate fractions are constants. 
 If a, b, Cy etc. are single roots, then ^=g' = r=---=l, and 
 
 F(x) x — a X — b X — c 
 
 The simpler fractions, into which the original fraction is thus 
 divided, may be integrated by means of the following formulas : 
 
 „ r hdx __ rhd(mx-\-n) _ h _^^ 
 
 J (mx -\- ny J m(mx -\- ny m(l — l)(mx -j- ny~^ i^ 
 
 „ek r hdx h . , 
 
 72. I — — loo: (mx-^n). 
 
 J mx-\-n m ^^ ^ 
 
 If any of the roots of the equation f(x) = are imaginary, the 
 parts of the integral which arise from conjugate roots can be com- 
 bined, and the integral thus brought into a real form. The following 
 formula, in which i = V— 1, is often useful in combining logarithms 
 of conjugate complex quantities : 
 
 73. log (x + yi) = log ?' + <^^, x = r cos <^, y = r sin <^. 
 
IRRATIONAL ALGEBRAIC FUNCTIONS 11 
 
 ^1| II. IRRATIONAL ALGEBRAIC FUNCTIONS 
 
 A. Expressions involving ^a-\-hx 
 
 The substitution of a new variable of integration, y = V a -f- bx, 
 gives 
 
 74. j-y/a -f bxdx = — V(a + bxf. 
 
 75. r.V^TS<^x=-2(2^^1M^5i±M. 
 J 15 b^ 
 
 ^ 105 b^ 
 
 77. rv£±jg^x=2v;^Tto+ar ,jf_ - 
 
 J X J X Va 4- ^ic 
 
 r__dx___ _ 2 V(x + ^i^ 
 J y/a-j-bx ^ 
 
 rxd^ 2(2a-bx) ^y-— 
 80. / , = -^^ TTTF^ Va + bx. 
 
 81. / 7= = — r^log -^=^= j=)' 
 
 Jx\a-{-bx Va \^a -\-bx -\-^a/ 
 
 . r J^ = ^tanh- J^±^, or ^ctnh- J^±:^. 
 J x^a-\-bx ■\la y a Va > a 
 
 82 
 
 ' + 
 
 , dx Va + bx b f* dx 
 
 83. 
 
 / dx _ _ Vq^ + bx b r di 
 
 x^ Va + bx <^^ 2a J xVa 
 
 ^bx 
 
12 IRRATIONAL ALGEBRAIC FUNCTIONS 
 
 4- to 
 c?a; _ _ V<^ + ^a; {2n — Z)h r dx 
 
 r dx _ _ Va + 6a; _ (^Zn—'S)}) r dx^ 
 
 J x« Va + to ~ (7i-l)aa;"-i (2n-2)aJ ^^-i-y/^ 
 
 n - w-2 
 
 88. I ^— -^- — ^^ = b j (a + bx) 2 dx + a j ^'-—^ — ^ d'o;. 
 
 ^ x(a-\-bxY J x(a + bx) ^ J { 
 
 + to 
 2 
 
 (a 4- to)2 c/ a;(o^ + to) 2 ^ (<^ + to)2 
 
 B. Expressions involving v a;^ ± d^ and Va^ — x^ 
 
 90. C-\/x'±a^dx = -^[ir Vic^ ± ci'^ ± a^ \og{x + V^2^~^)]-* 
 
 91. r Va^ - x^dx = i L Va^ - x^ + «' sin-i (^^1 • 
 
 92. r^=^ = log(x4-V^^T^^).^ 
 
 93. r /^ = sin-i(^-\ or - cos-^/^-). 
 
 94. / = -cos~M-)? or -sec~M-)- 
 
 95 
 96 
 
IRRATIONAL ALGEBRAIC FUNCTIONS 13 
 
 ^1 • 97. / dx = Vic^ — o? — a cos~^ - 
 
 J X X 
 
 ^^ . xdx 
 99 
 
 100. Cx -slx^ ± a^dx = ^ V(x2 ± ^2)8^ 
 
 101. fic Va^ - a^^t^a; = - ^ V(a2 _ x'f. 
 
 102. fV(^T^'dx 
 
 103. C-V^^F^^^^dx 
 J [. V(^^3^« + ^ V^^3T^ + ^ sin- g. 
 
 104. r_^=— ^£= 
 
 ^r^„ r dx X 
 
 105. / . = , 
 
 xdx — 1 
 
 
 107. 
 108 
 
 xdx 1 
 
 . fic V(ic2 _j_ ^2>)8^^ ^ ^ V(a;2 ± ay. 
 109. jx^(a^ - a^ydx = - ^ V(a2 _ x'f. 
 
 * See note on page 12 
 
14 IREATIONAL ALGEBRAIC FUNCTIONS 
 
 110. jx^^x" ± a^dx 
 
 111. ixWa^-a^dx 
 
 = -^-s/(a'-xy + ^(xV^'^^'+a'sin-^^' 
 
 
 113. r-^2^=-^v^rr^ + ^%m 
 
 
 114. / . .Ill— =T -'"^^"^ 
 
 2 ' a-iC 
 
 115. I — —== = ^ 
 
 ii/. I ^ o(ic = sm~^-- 
 
 J x^ X a 
 
 119. r_-£^=. = -^=_sin-iE. 
 
 C. Expressions involving Va + h^+c^ 
 
 4c 
 
 Let X = a + Z>ic + cx^, q = A:ac — h^, and A; = In order 
 
 to rationalize the function f{x, Va ■\-bx-\- cx^) we may put 
 Vo- 4- ftic + cx^ = V± c vCr + ^B^i^, according as c is positive 
 or negative, and then substitute for x a new variable z, such that 
 
 « See note on page 12 
 
IRRATIONAL ALGEBRAIC FUNCTIONS 
 
 15 
 
 z = V^ -\-Bx^-x^ — ic, if c > ; 
 
 z = 5 if c < and > 
 
 X — c 
 
 X — B 
 A f wliere a and yS are the roots of the equation 
 
 A -\- Bx - x^ = 0, ii c<0 SLud-^ <0. 
 
 — c 
 
 By rationalization, or by the aid of reduction formulas, may be ob- 
 tained the values of the following integrals : 
 
 or —rsinh-M— =4=), if c>0. 
 
 Vc \-s/4:ac-by 
 
 21. f -7=^ = -7= sm-M . - ),iic<0. 
 
 ^ 
 
 22 
 23 
 24 
 25, 
 
 dx _ 2(2cx + h) 
 XVX qVx 
 
 dx _ 2(2cx-\-b) /I 
 
 r dx 2{2cx-^-b) /l \ 
 
 •jx^Vx~ SqV^l Kx^^^r 
 
 r dx _ 2{2cx + b)-\fx 2 7c(n-l) C d 
 
 ' J X-^X~ {2n-l)qX- ^ 271-1 j A— 1 
 
 vz 
 
 dx 
 
 J 4(» + l)<; ^2{n + l)kJ 
 
 fxdx _ Vx b r dx 
 
 dx 
 
 Vx' 
 
 ' X'^dx 
 
6 IRRATIONAL ALGEBRAIC FUNCTIONS 
 
 / xdx __ 2{bx-\-2 a) 
 X^X~ q-y/X 
 
 / xdx Vx h r dx 
 
 XWX~~ (2n-l)cX'^~YcJ x-Vx' 
 
 ^^ rx'dx I X ^b\ ,- , 3Z»2_4ac r dx 
 
 r x^dx _ (2b''-4:ac)x-^2ab 1 r dx 
 J X-y/x~ cqVx cj VX 
 
 r x^dx _ (2b^-4:ac)x-^2ab 4.ac + (2n-S)b'' r dx 
 ' J X''y/X~ (2n-l)cqX"-Wx {2n-l)cq J x^-^Vx 
 
 rxHx (x" 5bx 5b^ ^^\VT (^^ ^^M r ^^ 
 ^^•JVz~V3c 12c2"^8c« 36-V Uc^ 16cVj Vx* 
 
 39 
 
 C xX'^dx _ X" Vx ^ r X"c?a; 
 
 •j VX "(271 + 1)0 2 J VX* 
 
 40 r^!^™^ ^ a;X"Vx _ (2n + 3)/> T^X"^ 
 •j Vx 2(71 + l)c 4(7i + l)cj Vx 
 
 ~2(7i + l)cj Vx* 
 4l.ja^VX(^a: V 8c ^48c2 3c/ 5o 
 
 (l$-S)/^- 
 
IRRATIONAL ALGEBRAIC FUNCTIONS 17 
 
 ^ 142. / — ^= plogi + — 7:=),ifa>0. 
 
 .^o r dx 1 . _,/ bx-\-2a \ 
 
 143. / 7== ; sm-M . ?ifa<0. 
 
 J a: VX V- a \a: V 6^ - 4 ac/ 
 
 144. / — = — , if a = 0. 
 
 J xVx ^^ 
 
 i C ^^ - v^ , 1 C dx b_ r dx 
 
 J xX"" Vx ~ (2 71 - 1) aX" G^ J ^x"-i Vx 2 aj a'» Vz 
 1 f dx _ Vx b r dx 
 
 145. 
 
 xVx 
 
 'y/Xdx rz: . h r dx . C dx 
 
 / X'^dx X" r x'^-Hx ^ r x^'-Hx 
 
 xVx~ {2n-l)-yfx ^J xVx 2J ^x ' 
 
 iAe% r^^dx Vx b r dx , T d^O! 
 
 /x'^dx _ 1 rjf^2^^dx_ _ ^ r x^'-^dx a P x'^-^dx 
 A^«Vx~ V j^«-iVa V z^Va V jt"VA 
 
 151 r^;!£!^ _ a^^-^A^Vx _ (2n + 2m-l)b C x'^-^X^dx 
 Ak * J Va ~" (2 7i + m)c 2c(2 7i4-m) J Va 
 
 {m — l)a r x'^-^X'^dx 
 (2n-\-m)cJ Va 
 
 152. ^ <"- ^ 
 
 'h 
 
 'A" Va (m — 1) ax*" - 1 A" 
 
 (2n+2m-S)b C dx 
 2a(m-l) J x'^-^X'^Vx 
 
 (2n-\-7n — 2)c r dx • 
 (m-l)a J x^-^X^-\/x' 
 
18 IREATIONAL ALGEBRAIC FUNCTIONS 
 
 153 r X"dx _ X-Wx (2n-l)h rX-2^ 
 ' J x^Vx (m-l)a^— 1 2(m-l) J ^m-iVA 
 
 . (2n-l)c r X^'-^dx 
 m-1 J X'^-^Vx 
 
 J (a' + b'x) Vx V- A 2 5' V- A A' 
 
 or JLin. 2A4-m(a^ + ^>^^)-2^>^VXx 
 
 where m = 5^' — 2 a'c and /?. = a^*'^ — aW + m'^. 
 
 If 7i = 0, the value of the integral is - 2 5' VA/[m(a'+ J'cc)]. 
 
 D. Miscellaneous Algebraic Expressions 
 
 155. / ■\/2 ax — x^ dx = ^l{x — a) V2 ax — x"" + a^ sin-i(aj — a)/a]. 
 
 156. r-=£= = cos-(^:if). 
 J ■\/2ax-x' \ a / 
 
 ,^^ C dx __2_, _, -h\a + hx) 
 
 157. / , , = / : tan-1 * ^ ^ 
 
 J -ya -\- hx . Va' + ^>'a:; a 
 
 2 ^ , _, I //(^ + hx) 
 or — 7=tanh \ ttVtttt 
 
 + ^»£C . Va' + ^»':z; V^^ ^^ N ^» (a' + h'x) 
 
 158. C^{a + ^'a^)(a' + b^x) dx = ^ "^ ^ ^^^^f "^ ^'^ V(a + hx){a^ + b'x) 
 'J -Ja^hx. ^a' -{-b^x 
 
 Sbb 
 159 
 
 r \ a'-hb'x _ Vg + bx • ^a' + h'x k r dx 
 
 J \a + bx '^~ b 2bJ ^a + bx-\/a'-{-b'x' 
 
 160. f\l^-^ dx = sin-la^ - Vl - x^ 
 
IRRATIONAL ALGEBRAIC FUNCTIONS 
 
 19 
 
 
 ■^/(x-a){a'-x) ya'-a 
 
 jgg r (px-\-q)dx ^ q-\-a'p P dx 
 
 J {x-a%x-h')^a + hx + cx' (^' - b' J (x-a')-\/a-{-bx-\-cx' 
 
 dx 
 
 q-\-b'p r 
 a' - b' J (x - b')-y/a -\- bx -^ ex" 
 
 164 
 
 cx^ 
 
 /dx 
 (a'-\-b'x)-\/a-^bx-{- 
 
 1 , / 2 ^ + m(a^ + b'x)-2b' •\lh{a -\-bx-\- c^ \ 
 
 or 
 where 
 
 "I^ \^2 V sf—hia + bx-^cx^)) 
 
 m = bb' ^2 a'c and h = a&'^ — a'bb' + ca 
 
 165. / , =--;\~i r-tan-iic\-7- — ; — ^j 
 
 J (a' + c'x^) Va + cx^ (^ ^ «^c - « ^^ ^ ^ («^ + ^» ) 
 
 ^ 
 
 or 
 
 a' ^ a'c — ac' Va + cx^ — x yJia'c — aG')/a' 
 
 C xdx 1 c^ ^ 1 ^ /^ (^ + ^*^ 
 166. I , =iA/~l :tan-iV-^^ fr 
 
 J_ I g' Va + Gx^--yl{ac' - 6^'c)/c' 
 
 2 c'^/ o^c' - a'c ^^ Va + cx^ + V(ac' - a'c)/c'' 
 
20 TRANSCENDENTAL FUNCTIONS 
 
 III. TRANSCENDENTAL FUNCTIONS 
 
 67. I sin xdx — — cos x. 
 
 68. I ^\Ti?xdx = — -J- cos X sin x -\- ix = \x — ^ sin 2 x. 
 
 69. I s,m^xdx = — -J cos ic (sin^ic -f 2). . 
 
 „n r ' 7 sin"-^a;cos£c , n — 1 T . „ , 
 
 70. I sin"£crfa7 = 1 I sin"-^^^^,^ 
 
 J n n J 
 
 71. I cosccc^ic = sinic. 
 
 72. r cos^a;c?ic = ^ sin cc cos cc + ^ a^ = ^ cc -|- j- sin 2 ic. 
 
 73. I cos^xdx — -J since (cosmic + 2). 
 
 74. I cos"icc?£c = - cos^'^ic sin a; H / (iO%*'~'^xdx. 
 
 J n n J 
 
 75. I sin X cos xdx = ^ sin^ic. 
 
 76. I sin^ic co^^xdx = — ^ (J sin 4 ic — £c). 
 
 »» r ' m ^ cos'«+ix 
 
 77. I sin X cos"*ic ace = — ^ • 
 
 »a r • m 1 sin'^+^ic 
 
 78. I sin"*ic cos xdx — — • 
 
 J ^ + 1 
 
 ffft r m • « 7 cos'^-^icsin^+iic , m — 1 r ^ „ • , 7 
 
 79. I cos"*cc sin**icaic = h I cos"*~^£c sm^icaa;. 
 
 J m -\- n m-h /ij 
 
 on r m • n J sin"-^iccos'»+^a! , 71 — 1 T ^ • „ 2 7 
 oO. I GOS"*xsin''xdx = 1 | cos'"a!Sin"^xaa:i 
 
 «- f eos'^xdx _ cos"'''"^a; m — n -\- 2 C co^'^xdx 
 
 J sin'a; (ti — l)sin"~^ic ti — 1 J sin''~'*a; 
 
TRANSCENDENTAL FUNCTIONS 21 
 
 ~^xdx 
 
 mL _ rcos^'xdx _ cos"*~^a; m — 1 rcos'^-^x 
 
 ^ J sin"ic (m — ?i) sin"~^ic m — nj sin^a; 
 
 sm' 
 
 (i-) 
 
 sin"'x cos"a; 
 
 n 
 
 J^ 1 m + 7i-2 r cga; 
 
 — 1 sin'^-^x • cos"~^a7 ti — 1 J sin"*ic • cos^-^a? 
 
 71-2 r 
 -1 js 
 
 1 1 7n-{-n — 2 f* dx 
 
 m — 1 sin"*~^a; • cos^'^a; w — 1 J sin"*' ^£c- cos" a;. 
 
 /-; — ^^^^^^ = log tan X. 
 sin a; cos a; 
 
 J.- r_dx^_ _ _ 1 cos a; m — 2 T 
 
 ' J sin'"a; m — 1 sin"'~^a7 m — ij si 
 
 sm"* "a; 
 
 QR r ^^ — ^ sina; n — 2 T c?a; 
 
 J cos^x 71 — 1 cos"-^a; n — lj cos^-^a; 
 
 87. I tan xdx =— log cos a;. 
 
 88. I tan^ ajc^aj = tana; — ar. 
 
 /tan**~^a! /* 
 tan"a;c?a; = t- — j t^n^'-^xdx. 
 
 90. I ctnajc^a; = log sina;. 
 
 91. I ctn^ajc^a; = — ctn x — x. 
 
 /ctn'*~^a; r 
 ctn"a'6?x = I ctn" -2 a; c^a;. 
 
 93. I seca;6?a; = logtan (-J + -)• 
 
 94. I sec^ajc^a; = tana;. 
 
22 TRANSCENDENTAL FUNCTIONS 
 
 195. lsec''xdx= I — ;; — 196. I cscsc^^a; = losr tan^a;. 
 
 197. I csc^jcc^ic =— etna?. 198. /csc"icc?x= / . ^ • 
 
 J J J sm"a: 
 
 199. / — r^ = — 7==tan-i — — ^— , 
 
 J a-{-bcosx -yJd^ _ ^2 a + 6 
 
 1 , V^^ — a^ tan hx-\-a-\-'b 
 
 . or ■ log , ^ — 3 
 
 t= -sly" -a? ^ ■\/lP--a^t^xv\x-a-h 
 
 ^ 2 , ^ , V^^H^tan^a; 
 
 ., or , tann~^ — ^— > 
 
 V V^^ - a2 a + ^> 
 
 t= / 
 
 I 2 , , , V ft^ — a^ tan -i- a; 
 
 ' or — ==ctnh-i — - — 
 
 200. ( ——J ; : 
 
 J a -\- cos X -\- c sin x 
 
 .(a — b) tan i x + c 
 
 , tan-^ ^^ , ^ ^ ^—: 
 
 Va2 _ J2 _ ^2 Va^ -b^-c" 
 
 t= 1 , (a - 5) tan 4a; + c - VZ»24- c^- a^ 
 x/ or , = log ^ , 
 
 V V^2_^c2-a2 ^(a_^,)tan|a; + c + V^2:^7-^ 
 
 V — 2 . 1 1 («^ — ^) tan 4- a? + c 
 
 ^ tanh-^ ^ , ^ ^ ^ — > 
 
 — 2 , , 1 (a — ^) tan 4- a? + c 
 
 V^TfTTT^ V62 + c^ - a^ 
 
 201. / x sin ajcZa? = sin x — x cos a?. 
 
 202. / x^ sin asc^a; = 2 a; sin a; — (a:*^ — 2) cos a;. 
 
 203. J x^sinxdx = (Sx^— 6) sin a; — (a;* — 6 a;) cos x. 
 
 204. I a:"* sin xdx = — ^'"cos x -\- m I x"*" ^cos a;c?x. 
 
TEANSCENDENTAL FUNCTIONS 23 
 
 ^H 205. / X cos xdx = cos x -[-x sin x. 
 
 206. I cc^cos xdx = 2 X Go^ X -{- (x^ — T) sin ic. 
 
 207. / cc'cos £C(^ic = (3 cc^ — 6) cos x -\-(x^ — 6 x) since. 
 
 208. I ic"* cos icc?x = cc"" sin a^ — m I o-"""^ since fZa;. 
 
 --- /*sinic , 1 since 1 /^coscc ^ 
 
 209. I dx= ' 7 H I 7 dx. 
 
 J x"" m — 1 £c"*-i m — IJ cc"*-' 
 
 «,- Tcoscc , 1 coscc 1 Tsincc , 
 
 210. I dx = 7 I r dx. 
 
 J x^ m — 1 cc"*~^ m — 1 J £c"'~^ 
 
 on r since _ cc^ x^ cc^ cr;^ 
 
 213. I sin (mx + a) • sin (nx + &) c?£c 
 
 _ sin (mx — nx -\- a — b) sin (mcc -\- nx -{- a -\- b) 
 
 2 (m — n) 2(m ■\- n) 
 
 214. I cos (mx + a) • cos (nx -\- b)dx 
 
 _ sin (mx ■\- nx ■\- a -\- b^ sin (mcc — rice -f- a — J) 
 
 ~ 2 (m 4- w) 2 (m - ti) 
 
 215. I sin (mx -\- a) • cos (nx -\- b)dx 
 
 _ cos (mx -\- nx -\- a -{- b) cos (ma? — nx -^ a —b) 
 ~'~ 2(m + n) ~ 2(m — n) 
 
 216. I sin (mcc + a) • sin (mcc + ^) c?cc 
 
 X ,, ^ sin (mcc + a) • cos (mcc + 5) 
 = - • cos (Z> - a) ^ ^ ^ ^• 
 
 2 ^ -^ 2m 
 
 217. I sin (mcc + a) • cos (mcc -^b)dx 
 
 sin (mcc -f a) •, sin (mcc + 5) cc . ,, 
 
 = ^ '- ^ ^ - - • sm (^ - a). 
 
 2m 2 ^ ^ 
 
24 TRANSCENDENTAL FUNCTIONS 
 
 218. / cos {mx H- a) • cos (mx + b)dx 
 
 X „ ^ sin (mx -h ci) cos (ttix + &) 
 
 = - • cos (^» - a) H ^ :^ ^ ^ 
 
 2 ^ ^ 2m 
 
 219. I sin~^ icc?a; = x siii~^ic + Vl — x^, 
 
 220. I cos~^icc?ic = X cos~^£c — Vl — x^. 
 
 221 
 222 
 
 I taii-^a;(^x = ic tan-^£c — ^ log (1 4- ar^. 
 I atn-'^xdx = x ctn-^a; + ^ log (1 + a;^. 
 
 223. / versin"^a;c?ic = (a; — 1) versin~^ic + V2ic — cc^. 
 
 224. C(sm-^xydx = ic (sin-icc)^ - 2 x + 2 Vl - cc^ gin-ia;. 
 
 225. fa; . siii-^xdx = ^ [(2 cc^ _ l) sin-iic -f a; Vl - x^]. 
 
 ^^^ r .17 x'^+^sin-ia; 1 r x^'+^dx 
 
 226. I a;"sin-ix<^a; = — ^ — r I ■ , 
 
 J n + 1 ^ + lJVl-ar* 
 
 _.^ r 1 7 a;"+icos-ia^ , 1 f a;"+^c?a^ 
 
 227. ./ a;"cos-ix^x = — j — + — — r | /:, - ' 
 
 J 7^ + 1 n-{-lJ Wl — x^ 
 
 ^^^ r . 17 a;"+Uan-ia; 1 Px^'+^dx 
 
 228. I a;" tan-i xc^a; = — j — r | . . » - 
 
 J 71 + 1 ^ + 1J 1 + ar^ 
 
 229. I logajc^a; = a; log a; — x. 
 
 230. r2^^^^x = -^aoga^)"+^ 
 
 J X 71 + 1 ^ ^ 
 
 231. j— ^^ = log(loga^). 
 
 dx 
 
 log a; 
 
 232 
 
 r dx 
 
 J x(\ogx) 
 
 :y (71-1) (log a;)"-i 
 
 233. f.T-loga^cZa;=:a;-+^r^^- , ^.J . 
 
 ^ 
 
TRANSCENDENTAL FUNCTIONS 25 
 
 SA 
 
 /" 
 
 234. I e'^dx = - 
 
 xe'^dx = — iax-l). 
 
 x'^e'^dx = I x"'-'^e"^dx. 
 
 a aj 
 
 c^^c C ^ 7 e"^ logic 1 re«^ , 
 
 238. I e«^logicc?a;= ^ | — dx. 
 
 J a aJ X 
 
 239. Je- . .inpxdx = ^-(^sin^^-^_^-|^cos^.) ^ 
 
 241. I sinh ajc?ic = cosh a;; i GO%h.xdx = sinhcc. 
 
 242. I tanh xdx = log cosh a? ; / ctnh xdx — log sinh ic. 
 
 243. rsech£c^a; = 2taii-i(e^). 
 
 244. I cschicc^a; = logtanh(-)' 
 
 245. I £c ^mh.xdx = « cosh x — sinh x. 
 
 246. I x cosh xdx = x sinh ic — cosh ic. 
 
 247. I cosh^a;c?a; = ^ (sinhx coshic + x). 
 
 248. I sinh x cosh £cc?a; = \ cosh (2 cc). 
 
 249. / sinh^ icc^ic = ^ (sinh a; cosh x — x). 
 
26 MISCELLANEOUS DEFINITE INTEGRALS 
 
 IV. MISCELLANEOUS DEFINITE INTEGRALS ^«g 
 
 250. j^*^:^ = |' if «>0; 0, if a = 0; -|, if a<0. 
 
 251. r ic''-ie-*c^x= r I log- I dx = T(n). 
 
 r(7i + 1) = n . r(7i), if 71 > 0. r(2) = r(i) = i 
 
 T(n-\-l) = nl, if 71 is an integer. r (^) = V^. 
 
 r (7.) = n (7. - 1). z (y) = i), [log r (t/)] 
 
 Z(l)=- 0.577216. 
 
 oco r' ™ 1/-. Nn 1^ r* x^'-'dx T(m)T(n) 
 
 Jo ^ Jo (l + ^r + '* r(7^ + 7i) 
 
 253. J sin"xe^ic = i cos'^xdx 
 Jo Jo 
 
 1 . 3 . 5 . . . (71 - 1) TT .- . 
 
 = — ^^ — - — ^ — , ^ • — > if 71 IS an even integer ; 
 
 ^ • 4 • O • • • ( 71-) ^ 
 
 2.4. 6-. .(7.-1) ... . . _ 
 
 = -q — ^ — z — ^7 5 II 71 IS an odd integer ; 
 
 1 • O . O • < ... 71 
 
 p/^ + l 
 
 1 /- V 2 / 
 = 71^'^ — 7 T for any value of 7i greater than —1. 
 
 rt_- /** sin77iccc?aj tt.. /x/^.n ^^ 7r._ 
 
 254. / 2 = 77' ifw>0; 0, if7?i = 0; --> if77i<0. 
 
 X 2 ' " '""^ " ' "' " '"^ ^ ' 2 
 
 nee T " sln OJ • COS 77107 C?iC ^ .„ 
 
 255. / = 0, if 771- < - 1 or 77i > 1 ; 
 
 Jo ^ 
 
 — > if7?i = — 1 or 771 = 1; — > if —l<m<l. 
 4 2 
 
 oKfi r* sin^ajc?a; tt . 
 
MISCELLANEOUS DEFINITE INTEGRALS 
 
 27 
 
 in 257. r"cos {x'')dx = rsin (x')dx = i ^ 
 
 TT 
 
 2* 
 
 258. 1 sin kx sin mxdx = / cos kx cos mxdx = 0, [7c =7^ m]. 
 
 259. / sin A;ic cos mxdx = -z ^> if A; — m is odd ; 
 
 Jo ^ - ^ 
 
 = 0, if A; — m is even. 
 
 260. 1 sm^Tnxdx^ 1 GOS^mxdx = '^' 
 
 261. / sin kx cos kxdx = 0. 
 
 
 263 
 
 264. 
 
 Jo G^ + ^cosa; vV_^ 
 
 Jf " cosmxdx TT . , .„ . ^ 
 
 Vl-A:^ 
 
 :^ = iir 
 
 sm'ic 
 
 Jr* cosa;c?a; _ T" sinxdx _ & 
 
 265. r 
 
 ^ '=fh©'»--(i-?--(SI)'--]'"- 
 
 IT 
 
 <1. 
 
 266. 
 
 267 
 268 
 
 Wl-k''sm''x'dx = i: 
 
 .jrv.-^.=i:5^=^,ifn>-i,«>o. 
 
28 MISCELLAKEOUS DEFINITE INTEGRALS 
 >' \ e "^ ^dx = Yy > if a > 0. 
 
 270. 
 
 271. I e-'^QQ^mxdx^-r-^ ^>ifa>0. 
 
 £ 
 £ 
 
 7n 
 272. / e-'^sin mxdx = -rr- z j if a > 0. 
 
 r -^ 
 
 «-.« r* 2 2 y -. V TT • e 4a2 
 
 273. I e-«^ COS bxdx = > if a > 0. 
 
 Jo 2a 
 
 274. rM^.z.=-^. 
 
 ^^^•i 1 + 0.^^- 12 
 
 280. jT'x- logg)"<fa = (£^|^> [» + 1 > 0, » + 1 > 0]. 
 
 ' log sin ictZic = I log COS iPc^x = — — • log 2. 
 
 Jo ^ 
 
 XT ^ 
 
 ic • log sin iccZic = — — log 2. 
 
 
TABLES 
 
 29 
 
 
 Natural Logarithms of Numbers between 1.0 and 9.9 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1. 
 
 0.000 
 
 0.095 
 
 0.182 
 
 0.262 
 
 0.336 
 
 0.405 
 
 0.470 
 
 0.531 
 
 0.588 
 
 0.642 
 
 2. 
 
 0.693 
 
 0.742 
 
 0.788 
 
 0.833 
 
 0.875 
 
 0.916 
 
 0.956 
 
 0.993 
 
 1.030 
 
 1.065 
 
 3. 
 
 1.099 
 
 1.131 
 
 1.163 
 
 1.194 
 
 1.224 
 
 1.253 
 
 1.281 
 
 1.308 
 
 1.335 
 
 1.361 
 
 4. 
 
 1.386 
 
 1.411 
 
 1.435 
 
 1.459 
 
 1.482 
 
 1.504 
 
 1.526 
 
 1.548 
 
 1.569 
 
 1.589 
 
 5. 
 
 1.609 
 
 1.629 
 
 1.649 
 
 1.668 
 
 1.686 
 
 1.705 
 
 1.723 
 
 1.740 
 
 1.758 
 
 1.775 
 
 6. 
 
 1.792 
 
 1.808 
 
 1.825 
 
 1.841 
 
 1.856 
 
 1.872 
 
 1.887 
 
 1.902 
 
 1.917 
 
 1.932 
 
 7. 
 
 1.946 
 
 1.960 
 
 1.974 
 
 1.988 
 
 2.001 
 
 2.015 
 
 2.028 
 
 2.041 
 
 2.054 
 
 2.067 
 
 8. 
 
 2.079 
 
 2.092 
 
 2.104 
 
 2.116 
 
 2.128 
 
 2.140 
 
 2.152 
 
 2.163 
 
 2.175 
 
 2.186 
 
 9. 
 
 2.197 
 
 2.208 
 
 2.21d 
 
 2.230 
 
 2.241 
 
 2.251 
 
 2.262 
 
 2.272 
 
 2.282 
 
 2.293 
 
 Natural Logarithms of Whole Numbers from 10 to 109 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 
 2.303 
 
 2.398 
 
 2.485 
 
 2.565 
 
 2.639 
 
 2.708 
 
 2.773 
 
 2.833 
 
 2.890 
 
 2.944 
 
 2 
 
 2.996 
 
 3.045 
 
 3.091 
 
 3.135 
 
 3.178 
 
 3.219 
 
 3.258 
 
 3.296 
 
 3.332 
 
 3.367 
 
 3 
 
 3.401 
 
 3.434 
 
 3.466 
 
 3.497 
 
 3.526 
 
 3.555 
 
 3.584 
 
 3.611 
 
 3.638 
 
 3.664 
 
 4 
 
 3.689 
 
 3.714 
 
 3.738 
 
 3.761 
 
 3.784 
 
 3.807 
 
 3.829 
 
 3.850 
 
 3.871 
 
 3.892 
 
 5 
 
 3.912 
 
 3.932 
 
 3.951 
 
 3.970 
 
 3.989 
 
 4.007 
 
 4.025 
 
 4.043 
 
 4.060 
 
 4.078 
 
 6 
 
 4.094 
 
 4.111 
 
 4.127 
 
 4.143 
 
 4.159 
 
 4.174 
 
 4.190 
 
 4.205 
 
 4.220 
 
 4.234 
 
 7 
 
 4.248 
 
 4.263 
 
 4.277 
 
 4.290 
 
 4.304 
 
 4.317 
 
 4.331 
 
 4.344 
 
 4.357 
 
 4.369 
 
 8 
 
 4.382 
 
 4.394 
 
 4.407 
 
 4.419 
 
 4.431 
 
 4.443 
 
 4.454 
 
 4.466 
 
 4.477 
 
 4.489 
 
 9 
 
 4.500 
 
 4.511 
 
 4.522 
 
 4.533 
 
 4.543 
 
 4.554 
 
 4.564 
 
 4.575 
 
 4.585 
 
 4.595 
 
 10 
 
 4.605 
 
 4.615 
 
 4.625 
 
 4.635 
 
 4.644 
 
 4.654 
 
 4.663 
 
 4.673 
 
 4.682 
 
 4.691 
 
 Values in Circular Measure of Angles which are given in 
 Degrees and Minutes 
 
 1' 
 
 0.0003 
 
 9' 
 
 0.0026 
 
 3° 
 
 0.0524 
 
 20° 
 
 0.3491 
 
 100° 
 
 1.7453 
 
 2' 
 
 0.0006 
 
 10' 
 
 0.0029 
 
 4° 
 
 0.0698 
 
 30° 
 
 0.5236 
 
 110° 
 
 1.9199 
 
 3' 
 
 0.0009 
 
 20' 
 
 0.0058 
 
 6° 
 
 0.0873 
 
 40° 
 
 0.6981 
 
 120° 
 
 2.0944 
 
 4' 
 
 0.0012 
 
 30' 
 
 0.0087 
 
 6° 
 
 0.1047 
 
 50° 
 
 0.8727 
 
 130° 
 
 2.2689 
 
 6' 
 
 0.0015 
 
 40' 
 
 0.0116 
 
 7° 
 
 0.1222 
 
 60° 
 
 1.0472 
 
 140° 
 
 2.4435 
 
 6' 
 
 0.0017 
 
 50' 
 
 0.0145 
 
 8° 
 
 0.1396 
 
 70° 
 
 1.2217 
 
 150° 
 
 2.6180 
 
 7' 
 
 0.0020 
 
 1' 
 
 0.0175 
 
 9° 
 
 0.1571 
 
 80° 
 
 1.3963 
 
 160° 
 
 2.7925 
 
 8' 
 
 0.0023 
 
 2' 
 
 0.0349 
 
 10" 
 
 0.1745 
 
 90° 
 
 1.5708 
 
 170° 
 
 2.9671 
 
30 
 
 TABLES 
 Natural Trigonometric Functions 
 
 Angle 
 
 Sin 
 
 Csc 
 
 Tan 
 
 Ctn 
 
 Sec 
 
 Cos 
 
 
 0° 
 
 0.000 
 
 GO 
 
 0.000 
 
 00 
 
 1.000 
 
 1.000 
 
 90° 
 
 1 
 
 0.017 
 
 57.30 
 
 0.017 
 
 57.29 
 
 1.000 
 
 1.000 
 
 89 
 
 2 
 
 0.035 
 
 28.65 
 
 0.035 
 
 28.64 
 
 1.001 
 
 0.999 
 
 88 
 
 3 
 
 0.052 
 
 19.11 
 
 0.052 
 
 19.08 
 
 1,001 
 
 0.999 
 
 87 
 
 4 
 
 0.070 
 
 14.34 
 
 0.070 
 
 14.30 
 
 1.002 
 
 0.998 
 
 86 
 
 6° 
 
 0.087 
 
 11.47 
 
 0.087 
 
 11.43 
 
 1.004 
 
 0.996 
 
 86° 
 
 6 
 
 0.105 
 
 9.567 
 
 0.105 
 
 9.514 
 
 1.006 
 
 0.995 
 
 84 
 
 7 
 
 0.122 
 
 8.206 
 
 0.123 
 
 8.144 
 
 1.008 
 
 0.993 
 
 83 
 
 8 
 
 0.139 
 
 7.185 
 
 0.141 
 
 7.115 
 
 1.010 
 
 0.990 
 
 82 
 
 9 
 
 0.156 
 
 6.392 
 
 0.158 
 
 6.314 
 
 1.012 
 
 0.988 
 
 81 
 
 10° 
 
 0.174 
 
 5.759 
 
 0.176 
 
 5.671 
 
 1.015 
 
 0.985 
 
 80° 
 
 11 
 
 0.191 
 
 5.241 
 
 0.194 
 
 5.145 
 
 1.019 
 
 0.982 
 
 79 
 
 12 
 
 0.208 
 
 4.810 
 
 0.213 
 
 4.705 
 
 1.022 
 
 0.978 
 
 78 
 
 13 
 
 0.225 
 
 4.445 
 
 0.231 
 
 4.331 
 
 1.026 
 
 0.974 
 
 77 
 
 14 
 
 0.242 
 
 4.134 
 
 0.249 
 
 4.011 
 
 1.031 
 
 0.970 
 
 76 
 
 16° 
 
 0.259 
 
 3.864 
 
 0.268 
 
 3.732 
 
 1.035 
 
 0.966 
 
 76° 
 
 16 
 
 0.276 
 
 3.628 
 
 0.287 
 
 3.487 
 
 1.040 
 
 0.961 
 
 74 
 
 17 
 
 0.292 
 
 3.420 
 
 0.306 
 
 3.271 
 
 1.046 
 
 0.956 
 
 73 
 
 18 
 
 0.309 
 
 3.236 
 
 0.325 
 
 3.078 
 
 1.051 
 
 0.951 
 
 72 
 
 19 
 
 0.326 
 
 3.072 
 
 0.344 
 
 2.904 
 
 1.058 
 
 0.946 
 
 71 
 
 20° 
 
 0.342 
 
 2.924 
 
 0.364 
 
 2.747 
 
 1.064 
 
 0.940 
 
 70° 
 
 21 
 
 0.358 
 
 2.790 
 
 0.384 
 
 2.605 
 
 1.071 
 
 0.934 
 
 69 
 
 22 
 
 0.375 
 
 2.669 
 
 0.404 
 
 2.475 
 
 1.079 
 
 0.927 
 
 68 
 
 23 
 
 0.391 
 
 2.559 
 
 0.424 
 
 2.356 
 
 1.086 
 
 0.921 
 
 67 
 
 24 
 
 0.407 
 
 2.459 
 
 0.445 
 
 2.246 
 
 1.095 
 
 0.914 
 
 66 
 
 26° 
 
 0.423 
 
 2.366 
 
 0.466 
 
 2.145 
 
 1.103 
 
 0.906 
 
 66° 
 
 26 
 
 0.438 
 
 2.281 
 
 0.488 
 
 2.050 
 
 1.113 
 
 0.899 
 
 64 
 
 27 
 
 0.454 
 
 2.203 
 
 0.510 
 
 1.963 
 
 1.122 
 
 0.891 
 
 63 
 
 28 
 
 0.469 
 
 2.130 
 
 0.532 
 
 1.881 
 
 1.133 
 
 0.883 
 
 62 
 
 29 
 
 0.485 
 
 2.063 
 
 0.554 
 
 1.804 
 
 1.143 
 
 0.875 
 
 61 
 
 30° 
 
 0.500 
 
 2.000 
 
 0.577 
 
 1.732 
 
 1.155 
 
 0.866 
 
 60° 
 
 31 
 
 0.515 
 
 1.942 
 
 0.601 
 
 1.664 
 
 1.167 
 
 0.857 
 
 69 
 
 32 
 
 0.530 
 
 1.887 
 
 0.625 
 
 1.600 
 
 1.179 
 
 0.848 
 
 68 . 
 
 33 
 
 0.545 
 
 1.836 
 
 0.649 
 
 1.540 
 
 1.192 
 
 0.839 
 
 67 i 
 
 34 
 
 0.559 
 
 1.788 
 
 0.675 
 
 1.483 
 
 1.206 
 
 0.829 
 
 66 
 
 36° 
 
 0.574 
 
 1.743 
 
 0.700 
 
 1.428 
 
 1.221 
 
 0.819 
 
 65° 
 
 36 
 
 0.588 
 
 1.701 
 
 0.727 
 
 1.376 
 
 1.236 
 
 0.809 
 
 64 
 
 37 
 
 0.602 
 
 1.662 
 
 0.754 
 
 1.327 
 
 1.252 
 
 0.799 
 
 63 
 
 38 
 
 0.616 
 
 1.624 
 
 0.781 
 
 1.280 
 
 1.269 
 
 0.788 
 
 62 
 
 39 
 
 0.629 
 
 1.589 
 
 0.810 
 
 1.235 
 
 1.287 
 
 0.777 
 
 61 
 
 40° 
 
 0.643 
 
 1.556 
 
 0.839 
 
 1.192 
 
 1.305 
 
 0.766 
 
 60° 
 
 41 
 
 0.656 
 
 1.524 
 
 0.869 
 
 1.150 
 
 1.325 
 
 0.755 
 
 49 
 
 42 
 
 0.669 
 
 1.494 
 
 0.900 
 
 1.111 
 
 1.346 
 
 0.743 
 
 48 
 
 43 
 
 0.682 
 
 1.466 
 
 0.933 
 
 1.072 
 
 1.367 
 
 0.731 
 
 47 
 
 44 
 
 0.695 
 
 1.440 
 
 0.966 
 
 1.036 
 
 1.390 
 
 0.719 
 
 46 
 
 46° 
 
 0.707 
 
 1.414 
 
 1.000 
 
 1.000 
 
 1.414 
 
 0.707 
 
 46° 
 
 
 Cos 
 
 Sec 
 
 Ctn 
 
 Tan 
 
 Csc 
 
 Sin 
 
 Angle 
 
TABLES 
 
 31 
 
 Values of the Complete Elliptic Integrals, K and £, for Different 
 Values of the Modulus, k 
 
 -=f: 
 
 dz 
 
 Vl-A:2 
 
 sm-'z 
 
 
 sin-U- 
 
 K 
 
 E 
 
 sin-U- 
 
 K 
 
 E 
 
 sin-U- 
 
 K 
 
 E 
 
 0° 
 
 1.5708 
 
 1.5708 
 
 60° 
 
 1.9356 
 
 1.3055 
 
 81.0° 
 
 3.2553 
 
 1.0338 
 
 1° 
 
 1.5709 
 
 1.5707 
 
 61° 
 
 1.9539 
 
 1.2963 
 
 81.2° 
 
 3.2771 
 
 1.0326 
 
 2° 
 
 1.5713 
 
 1.5703 
 
 62° 
 
 1.9729 
 
 1.2870 
 
 81.4° 
 
 3.2995 
 
 1.0313 
 
 3° 
 
 1.5719 
 
 1.5697 
 
 63° 
 
 1.9927 
 
 1.2776 
 
 81.6° 
 
 3.3223 
 
 1.0302 
 
 40 
 
 1.5727 
 
 1.5689 
 
 64° 
 
 2.0133 
 
 1.2681 
 
 81.8° 
 
 3.3458 
 
 1.0290 
 
 6° 
 
 1.5738 
 
 1.5678 
 
 66° 
 
 2.0347 
 
 1.2587 
 
 82.0° 
 
 3.3699 
 
 1.0278 
 
 6° 
 
 1.5711 
 
 1.5665 
 
 66° 
 
 2.0571 
 
 1.2492 
 
 82.2° 
 
 3.3946 
 
 1.0267 
 
 70 
 
 1.5767 
 
 1.5649 
 
 67° 
 
 2.0804 
 
 1.2397 
 
 82.4° 
 
 3.4199 
 
 1.0256 
 
 8° 
 
 1.5785 
 
 1.5632 
 
 68° 
 
 2.1047 
 
 1.2301 
 
 82.6° 
 
 3.4460 
 
 1.0245 
 
 9° 
 
 1.5805 
 
 1.5611 
 
 69° 
 
 2.1300 
 
 1.2206 
 
 82.8° 
 
 3.4728 
 
 1.0234 
 
 10° 
 
 1.5828 
 
 1.5589 
 
 60° 
 
 2.1565 
 
 1.2111 
 
 83.0° 
 
 3.5004 
 
 1.0223 
 
 11° 
 
 1.5854 
 
 1.5564 
 
 61° 
 
 2.1842 
 
 1.2015 
 
 83.2° 
 
 3.5288 
 
 1.0213 
 
 12° 
 
 1.5882 
 
 1.5537 
 
 62° 
 
 2.2132 
 
 1.1921 
 
 83.4° 
 
 3.5581 
 
 1.0202 
 
 13° 
 
 1.5913 
 
 1.5507 
 
 63° 
 
 2.2435 
 
 1.1826 
 
 83.6° 
 
 3.5884 
 
 1.0192 
 
 14° 
 
 1.5946 
 
 1.5476 
 
 64° 
 
 2.2754 
 
 1.1732 
 
 83.8° 
 
 3.6196 
 
 1.0182 
 
 16° 
 
 1.5981 
 
 1.5442 
 
 66° 
 
 2.3088 
 
 1.1638 
 
 84.0° 
 
 3.6519 
 
 1.0172 
 
 16° 
 
 1.6020 
 
 1.5405 
 
 66.6° 
 
 2.3261 
 
 1.1592 
 
 84.2° 
 
 3.6853 
 
 1.0163 
 
 17° 
 
 1.6061 
 
 1.5367 
 
 66.0° 
 
 2.3439 
 
 1.1546 
 
 84.4° 
 
 3.7198 
 
 1.0153 
 
 18° 
 
 1.6105 
 
 1.5326 
 
 66.6° 
 
 2.3622 
 
 1.1499 
 
 84.6° 
 
 3.7557 
 
 1.0144 
 
 19° 
 
 1.6151 
 
 1.5283 
 
 67.0° 
 
 2.3809 
 
 1.1454 
 
 84.8° 
 
 3.7930 
 
 1.0135 
 
 20° 
 
 1.6200 
 
 1.5238 
 
 67.6° 
 
 2.4001 
 
 1.1408 
 
 86.0° 
 
 3.8317 
 
 1.0127 
 
 21° 
 
 1.6252 
 
 1.5191 
 
 68.0° 
 
 2.4198 
 
 1.1.362 
 
 86.2° 
 
 3.8721 
 
 1.0118 
 
 22° 
 
 1.63b7 
 
 1.5141 
 
 68.6° 
 
 2.4401 
 
 1.1317 
 
 86.4° 
 
 3.9142 
 
 1.0110 
 
 23° 
 
 1.6365 
 
 1.5090 
 
 69.0° 
 
 2.4610 
 
 1.1273 
 
 86.6° 
 
 3.9583 
 
 1.0102 
 
 24° 
 
 1.6426 
 
 1.5037 
 
 69.6° 
 
 2.4825 
 
 1.1228 
 
 86.8° 
 
 4.0044 
 
 1.0094 
 
 26° 
 
 1.6490 
 
 1.4981 
 
 70.0° 
 
 2.5046 
 
 1.1184 
 
 86.0° 
 
 4.0528 
 
 1.0087 
 
 26° 
 
 1.6557 
 
 1.4924 
 
 70.6° 
 
 2.5273 
 
 1.1140 
 
 86.2° 
 
 4.1037 
 
 1.0079 
 
 27° 
 
 1.6627 
 
 1.4864 
 
 71.0° 
 
 2.5507 
 
 1.1096 
 
 86.4° 
 
 4.1574 
 
 1.0072 
 
 28° 
 
 1.6701 
 
 1.4803 
 
 71.6° 
 
 2.5749 
 
 1.1053 
 
 86.6° 
 
 4.2142 
 
 1.0065 
 
 29° 
 
 1.6777 
 
 1.4740 
 
 72.0° 
 
 2.5998 
 
 1.1011 
 
 86.8° 
 
 4.2744 
 
 1.0059 
 
 30° 
 
 1.6858 
 
 1.4675 
 
 72.6° 
 
 2.6256 
 
 1.0968 
 
 87.0° 
 
 4.3387 
 
 1.0053 
 
 31° 
 
 1.6941 
 
 1.4608 
 
 73.0° 
 
 2.6521 
 
 1.0927 
 
 87.2° 
 
 4.4073 
 
 1.0047 
 
 1 32° 
 
 1.7028 
 
 1.4539 
 
 73.6° 
 
 2.6796 
 
 1.0885 
 
 87.4° 
 
 4.4812 
 
 1.0041 
 
 f 33° 
 
 1.7119 
 
 1.4469 
 
 74.0° 
 
 2.7081 
 
 1.0844 
 
 87.6° 
 
 4.5619 
 
 1.0036 
 
 ' 34° 
 
 1.7214 
 
 1.4397 
 
 74.6° 
 
 2.7375 
 
 1.0804 
 
 87.8° 
 
 4.6477 
 
 1.0031 
 
 36° 
 
 1.7312 
 
 1.4323 
 
 76.0° 
 
 2.7681 
 
 1.0764 
 
 88.0° 
 
 4.7427 
 
 1.0026 
 
 36° 
 
 1.7415 
 
 1.4248 
 
 76.6° 
 
 2.7998 
 
 1.0725 
 
 88.2° 
 
 4.8479 
 
 1.0022 
 
 37° 
 
 1.7522 
 
 1.4171 
 
 76.0° 
 
 2.8327 
 
 1.0686 
 
 88.4° 
 
 4.9654 
 
 1.0017 
 
 38° 
 
 1.7633 
 
 1.4092 
 
 76.6° 
 
 2.8669 
 
 1.0648 
 
 88.6° 
 
 5.0988 
 
 1.0014 
 
 39° 
 
 1.7748 
 
 1.4013 
 
 77.0° 
 
 2.9026 
 
 1.0611 
 
 88.8° 
 
 5.2527 
 
 1.0010 
 
 40° 
 
 1.7868 
 
 1.3931 
 
 77.6° 
 
 2.9397 
 
 1.0574 
 
 89.0° 
 
 5.4349 
 
 1.0008 
 
 41° 
 
 1.7992 
 
 1.3849 
 
 78.0° 
 
 2.9786 
 
 1.0538 
 
 89.1° 
 
 5.5402 
 
 1.0006 
 
 42° 
 
 1.8122 
 
 1.3765 
 
 78.6° 
 
 3.0192 
 
 1.0502 
 
 89.2° 
 
 5.6579 
 
 1.0005 
 
 43° 
 
 1.8256 
 
 1.3680 
 
 79.0° 
 
 3.0617 
 
 1.0468 
 
 89.3° 
 
 5.7914 
 
 1.0005 
 
 44° 
 
 1.8396 
 
 1.3594 
 
 79.6° 
 
 3.1064 
 
 1.0434 
 
 89.4° 
 
 5.9455 
 
 1.0003 
 
 46° 
 
 1.8541 
 
 1.3506 
 
 80.0° 
 
 3.1534 
 
 1.0401 
 
 89.6° 
 
 6.1278 
 
 1.0002 
 
 46° 
 
 1.8691 
 
 1.3418 
 
 80.2° 
 
 3.1729 
 
 1.0388 
 
 89.6° 
 
 6.3504 
 
 1.0001 
 
 47° 
 
 1.8848 
 
 1.3329 
 
 80.4° 
 
 3.1928 
 
 1.0375 
 
 89.7° 
 
 6.6385 
 
 1.0001 
 
 48° 
 
 1.9011 
 
 1.3238 
 
 80.6° 
 
 3.2132 
 
 1.0363 
 
 89.8° 
 
 7.0440 
 
 1.0000 
 
 49° 
 
 1.9180 
 
 1.3147 
 
 80.8° 
 
 3.2340 
 
 1.0350 
 
 89.9° 
 
 7.7371 
 
 1.0000 
 
32 TABLES 
 
 Common Logarithms of T{n) for Values of n between 1 and 2 
 
 t 
 
 n 
 
 log,or(n) 
 
 n 
 
 iogior(«) 
 
 n 
 
 log,or(n) 
 
 n 
 
 iog,or(7i) 
 
 n 
 
 log,or(n) 
 
 1.01 
 
 1.9975 
 
 1.21 
 
 1.9617 
 
 1.41 
 
 T.9478 
 
 1.61 
 
 1.9517 
 
 1.81 
 
 1.9704 
 
 1.02 
 
 1.9951 
 
 1.22 
 
 1.9605 
 
 1.42 
 
 1.9476 
 
 1.62 
 
 T.9523 
 
 1.82 
 
 1.9717 
 
 1.03 
 
 T.9928 
 
 1.23 
 
 1.9594 
 
 1.43 
 
 1.9475 
 
 1.63 
 
 1.9529 
 
 1.83 
 
 1.9730 
 
 1.04 
 
 1.9905 
 
 1.24 
 
 1.9583 
 
 1.44 
 
 1.9473 
 
 1.64 
 
 1.9536 
 
 1.84 
 
 1.9743 
 
 1.05 
 
 1.9883 
 
 1,25 
 
 1.9573 
 
 1.45 
 
 1.9473 
 
 1.65 
 
 1.9543 
 
 1.85 
 
 1.9757 
 
 1.06 
 
 1.9862 
 
 1.26 
 
 1.9564 
 
 1.46 
 
 1.9472 
 
 1.66 
 
 1.9550 
 
 1.86 
 
 1.9771 
 
 1.07 
 
 1.9841 
 
 1.27 
 
 1.9554 
 
 1.47 
 
 1.9473 
 
 1.67 
 
 1.9558 
 
 1.87 
 
 1.9786 
 
 1.08 
 
 1.9821 
 
 1.28 
 
 1.9546 
 
 1.48 
 
 1.9473 
 
 1.68 
 
 1.9566 
 
 1.88 
 
 1.9800 
 
 1.09 
 
 1.9802 
 
 1.29 
 
 1.9538 
 
 1.49 
 
 T.9474 
 
 1.69 
 
 1.9575 
 
 1.89 
 
 1.9815 
 
 1.10 
 
 1.9783 
 
 1.30 
 
 1.9530 
 
 1.50 
 
 1.9475 
 
 1.70 
 
 T.9584 
 
 1.90 
 
 1.9831 
 
 1.11 
 
 1.9765 
 
 1.31 
 
 1.9523 
 
 1.51 
 
 1.9477 
 
 1.71 
 
 1.9593 
 
 1.91 
 
 1.9846 
 
 1.12 
 
 1.9748 
 
 1.32 
 
 1.9516 
 
 1.52 
 
 1.9479 
 
 1.72 
 
 1.9603 
 
 1.92 
 
 1.9862 
 
 1.13 
 
 1.9731 
 
 1.33 
 
 1.9510 
 
 1.53 
 
 1.9482 
 
 1.73 
 
 1.9613 
 
 1.93 
 
 1.9878 
 
 1.14 
 
 1.9715 
 
 1.34 
 
 1.9505 
 
 1.54 
 
 T.9485 
 
 1.74 
 
 1.9623 
 
 1.94 
 
 1.9895 
 
 1.15 
 
 1.9699 
 
 1.35 
 
 1.9500 
 
 1.55 
 
 1.9488 
 
 1.75 
 
 1.9633 
 
 1.95 
 
 1.9912 
 
 1.16 
 
 1.9684 
 
 1.36 
 
 1.9495 
 
 1.56 
 
 1.9492 
 
 1.76 
 
 1.9644 
 
 1.96 
 
 1.9929 
 
 1.17 
 
 1.9669 
 
 1.37 
 
 1.9491 
 
 1.57 
 
 1.9496 
 
 1.77 
 
 1.9656 
 
 1.97 
 
 1.9946 
 
 1.18 
 
 1.9655 
 
 1.38 
 
 1.9487 
 
 1.58 
 
 1.9501 
 
 1.78 
 
 1.9667 
 
 1.98 
 
 T.9964 
 
 1.19 
 
 1.9642 
 
 1.39 
 
 1.9483 
 
 1.59 
 
 1.9506 
 
 1.79 
 
 1.9679 
 
 1.99 
 
 1.9982 
 
 1.20 
 
 1.9629 
 
 1.40 
 
 1.9481 
 
 1.60 
 
 1.9511 
 
 1.80 
 
 1.9691 
 
 2.00 
 
 0.0000 
 
 '4 
 
 rr(5! + i) = 3.r(z), if «>o; r(2) = r(i) = in 
 
 \ [r (x) ' r(l - X)] = TT/sin jra;, if 1 > a^ > 0. J 
 
 If the values of an analytic function, f{x), are given in a table for consecu- 
 tive values of the argument, x, with the constant interval d, and if h = kd, 
 where k is any desired fraction, 
 
 /(« + .)=/(„) + . .A, + *-<^.A, + *(^:i^ii^.A3 + .... 
 
 where /(a) is any tabulated value. 
 
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