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Home , Antiderivative, Calculus, Constant of integration, Derivative, Second derivative

-~.I--­ Appendix: TABLE OF

148. In the following table, the of integration, C, is omitted but should be added to the result of every integration. The letter x represents any ; u represents any of x; the remaining letters represent arbitrary constants, unless otherwise indicated; all angles are in . Unless otherwise mentioned, loge u = log u.

Expressions Containing ax + b

1. I(ax+ b)ndx= 1 (ax+ b)n+l, n=l=-1. a(n + 1) dx 1 2. = -loge

5. Ix(ax + b)n dx = 1 (ax + b)n+2 a2 (n + 2) b ---,,---(ax + b)n+l n =1= -1, -2. a2 (n + 1) , xdx x b 6. = - - -2 log(ax + b). I ax+ b a a

7 I xdx = b 1 I ( b) • (ax + b)2 a2(ax + b) + -;}i og ax + .

369 370 • Appendix: Table of Integrals

8 I xdx = b _ 1 • (ax + b)3 2a2(ax + b)2 a2(ax + b) . 1 (ax+ b)n+3 (ax+ b)n+2 9. x2(ax + b)n dx = - - 2b--'------'-- I a2 n + 3 n + 2 2 b (ax + b) n+ 1 ) + , ni=-I,-2,-3. n+l 2 10. I x dx = ~ (.!..(ax + b)2 - 2b(ax + b) + b2 10g(ax + b)). ax+ b a 2 2 2 11. I x dx 2 =~(ax+ b) - 2blog(ax+ b) _ b ). (ax + b) a ax + b 2 2 12 I x dx = _1(10 (ax + b) + 2b _ b ) • (ax + b) 3 a3 g ax + b 2(ax + b) 2 .

13. I xm(ax + b)n dx

= 1 (xm(ax+ b)n+l - mbIxm-1(ax+ b)n dx) a(m + n + 1) = 1 (xm+1(ax+ b)n+ nbfxm(ax+ b)n-l dX), m+n+l m> 0, m + n + 1 i= O.

14 I dx =.!..1 ( x ) • x(ax + b) b og ax + b .

15 I dx = _l- + ...E.. 10 (ax + b) • x2(ax + b) bx b2 g x . 2 16 I dx = 2ax - b ~ 1 ( x ) • x3(ax + b) 2b2x2 + b3 og ax + b .

17 I dx = 1 _ -..!..- 1 ( ax + b) • x(ax + b)2 b(ax + b) b2 og x .

18 I dx = l-[.!..( ax + 2b)2 + 10 ( x )] • x(ax + b)3 b3 2 ax + b g ax + b .

19 I dx = _ b + 2 ax 2a 1 (ax + b) • x 2(ax + b)2 b2x(ax + b) + b3 og x .

20. Iv'ax + b dx = ~v' (ax + b)3. 3a ~ ;--;--; 2(3ax - 2b) 21. xvax+bdx= 2 Y(ax+b)3. I 15a 2(l5a2x2 - 12abx + 8b2)v'(ax + b)3 22. x2v'ax + b dx = ---'------=----'--"'----'---- I 105a3 2(35a3x3 - 30a2bx2 + 24ab2x - 16b3)v'(ax + b)3 23. x3v'ax + b dx = --'------'------'----'-- I 315a4 • Appendix: Table ofIntegrals 371

24. J xmJax + bdx= a~1 Ju2(u2 - b)ndu, u= Yax+ b.

~ ~ J dx 25. J X dx= 2 ax+ b+ b xYax+ b'

26. J dx = 2~. Yax+ b a

27. J xdx = 2(ax- 2b) Yax+ b. Vax + b 3a2 x2 dx _ 2(3a2x2 - 4abx + 8b2) 28. J Yax + b - 15a3 Yax + b. x3 dx _ 2(5a3x3 - 6a2bx2 + 8ab2x - 16b3) 29. J Yax + b - 35a4 Yax + b. xn 30. Jy dx = J(u2 - b)n du, u = Vax + b. ax + b an:1

dx 1 (~-Vb) 31. JxYax + b = Vb log Yax + b + Vb for b> O.

32 J dx - 2 -1 J ax + b b < 0 • xYax+b-\7=btan -b'

_ -2 h-1Jax+ b -\7btan -b ' b>O.

33 J dx = Yax + b - -.!!:.-J dx · x2Yax + b bx 2b xYax + b' 2 34 J dx = _ Yax + b + 3a~ + 3a J dx • x3Yax + b 2bx2 4b2x 8b2 xYax + b'

35. J dx = 1 J (u - a)m+n-2 du ax+ b u= xn(ax+ b)m bm+n- 1 um ' x

36. J(ax + b)~n/2 dx = 2(ax + b)(2~n)/2. a(2 ± n) 2 (ax+ b)(4~n)/2 b(ax+ b)(2.~n)/2) 37. x(ax + b)~n/2 dx = - - . J a2 4 ± n 2 ± n

38. J dx = l..J dx - !!:. J dx . x(ax + b)n/2 b x(ax + b)(n-2)/2 b (ax + b)n/2 m 1 39 J xmdx = 2xmyax + b _ 2mb J x - dx • Yax+ b (2m+ l)a (2m+ l)a Yax+ b'

40. J dx = -Vax + b _ (2n - 3)a J dx . xnYax + b (n - 1)bxn- 1 (2n - 2)b xn- 1Vax + b

41. J (ax +xb)n/2 dx = a J (ax + b) (n-2)/2 dx + b J (ax + b~(n-2)/2 dx. 372 • Appendix: Table of Integrals

42. J dx = 1 10 (eX + d) be - ad O. (ax + b)(ex + d) be - ad g ax + b ' *

43 J dx · (ax + b)2(ex + d)

= 1 [ 1 + e 10 (ex + d)] be - ad o. be - ad ax + b be - ad g ax + b ' *

44. J(ax+ b)n(ex+ d)mdx= 1 [(ax+ b)n+l(ex+ d)m (m+ n+ 1)a

- m(be - ad) J(ax + b)n(ex + d)m-I dxJ

45. J dx = 1 ( 1 (ax+ b)n(ex+ d)m (m-1)(be- ad) (ax+ b)n-I(ex+ d)m-I

- a(m+ n- 2) J (ax+ b)n~:x+ d)m-I). m> 1, n> 0, be- ad* O.

46. J (ax + b)n dx (ex + d)m 1 ((ax+ b)n+1 J (ax+ b)ndX ) = - + (m - n - 2) a (m - 1) (be - ad) (ex + d) m- 1 (ex + d) m-1

= 1 ((ax+ b)n + n(be- ad) J (ax+ b)n-l dX). (m - n - l)e (ex + d)m-l (ex + d)m

47. J x dx = 1 (! In lax + ~ (ax + b)(ex + d) be - ad a

- ~ log(ex+ d»). be - ad * o.

48 J xdx 1 ( b · (ax+ b)2(ex+ d) = be- ad - a(ax+ b)

_ d 10 (ex + d)) be - ad be - ad g ax + b ' * o. ex+d 2 ~~ 49. J Yax + b dx = 3a2 (3ad - 2be + aex) V ax + b. 50.J~dx=2~ ex + d e

2 J ad - be - I e( ax b) -- tan J + e> 0, ad> be. e e ad - be ' 51.J~dx=2~ ex + d e

+! Jbe- ad 10 (Ye(ax+ b) - Ybe- ad) e> 0, be> ad. e e g Y e( ax + b) + Y be - ad '

52 J dx - 2 -I Je(ax + b) e> 0, ad> be. • (ex + d)Yax + b - VcYad - be tan ad - be ' • Appendix: Table ofIntegrals 373

53 f dx • (ex+d)Yax+b

= 1 10 (Y e(ax + b) - Ybe - ad) e> 0, be> ad. V"cY be - ad g Ye( ax + b) + Ybe - ad '

Expressions Containing ax2 + c, axn + c, x 2 ± p2, and p2 - x 2 dx 1 x ~ 1 54. fp2 + x2 = p tan-1 p or - etn- (;}

55 dx = _1 1 (~) ~ 1 • fp2 _ x2 2p og P- x ' or tanh- (;}

56. = ~ ~ tan- 1(x ~), a and e> O. fax:x+ e v ae -V -;; x~ ~) 57 f dx = 1 1 ( - n> 0, e< 0 • ax2 + e 2Y- ae og x~ + ~ , x~) = 1 1og(V"c+ a < 0, e> O. 2Y-ae V"c- x~ , 58. f dx = 1 x (ax2 + e)n 2(n - 1)e (ax2 + e)n-1 2n - 3 dx f n a positive integer. + 2(n - 1)e (ax2 + e)n-1' 1 (ax2 + e)n+1 59. x(ax2 + e) n dx = -2 ,n=F -1. f a n + 1 x 1 60. x2 dx = -log(ax2 + e). fa + e 2a 2 61 dx = 1.. 1 ( ax ) • fx(ax2 + e) 2e og ax2 + e .

62 f dx = _1.. - !:f dx • x2(ax2 + e) ex e ax2 + e' 2 63 f x dx = ~ - !- f dx • ax2 + e a a ax2 + e' xn dx xn- 1 e x n- 2 dx 64 = --f n =F 1. • fax2 + e a( n - 1) a ax2 + e ' 2 65 f x dx = 1 x • (ax2 + e)n 2(n - 1)a (ax2 + e)n-1 1 f dx + 2(n - 1)a (ax2 + e)n-1 . 374 • Appendix: Table of Integrals

66. I dx = 1- I dx - !!:... I dx . x2(ax2 + c)n C x2(ax2 + c)n-l c (ax2 + c)n

67. IVx2 ±p2 dx= ~[XVX2±P2±p2Iog(x+Vx2±P2)].

68. IVp2- x2 dx= ~[XVp2_X2+p2sin-l(;)l

dx - I( y' 2 + 2) 69·VI X2 ± p2 - og X + X - P. p:~ Sin-1C~) 1 70. IV X2 = or - cos- (;}

71. IVax2 + cdx= ;Vax2 + c + ~cr log(x~ + Vax2 + c), a> O. 2va a 72. IVax2 + cdx= ;Vax2 + c+ 2J-a Sin-1(xJ-c ). a< O.

2 73. I Va~ + c = ~ log(x~+ Vax + c), a> O.

1 a 74. I Va~ + c = ~ sin- (xJ-c ). a< O.

75. IxVax2 + c dx = _I_(ax2 + c) 3/2. 3a

76. Ix2Vax2 + c dx = ~V (aX2 + c)3 - E.-Vax2 + c 4a 8a c2 - ~Iog(x~+ y;;;2 + c), a> O.

77. Ix2Vax2 + c dx = ~V (ax2 + c)3 - E.-Vax2 + c 4a 8a 2 - sin-1( X- a< O. 8aV-ac pa)c '

xdx - IV 2 78. I Vax2 + c - -;; ax + c. x2 dx x 1 I 79. = - y'ax2 + c -- y'ax2 + c dx. I Vax2 + c a a 2 Vax + c d y' 2 ~ r I( VaX2 + c - ~) 80• x = ax + c + v c og , c>O. I x x 2 Vax + c d _ y' X2 ~ ~ -1 VaX2 + c 81· x- a +c- v-ctan ~~, c< O. I x v-c 2 dx __1- ( p + V p2 ± x ) 82. I xVp2 ± x2 - P log x . • Appendix: Table ofIntegrals 375

xndx xn- 1Yax2 +c (n-1)cf xn- 2 dx 87 = - --'------"-- n> O. . f Yax2 +c na na Yax2 +c' n 1 2 88. fxnYax2 + c dx = x - (ax + c) 3/2 (n + 2) a

_ (n - l)c fxn-2Yax2 + c dx n> O. (n + 2)a ' 2 2 89. fYax + c dx = _ (ax + c) 3/2 Xn c(n - 1)xn - 1 ~~- (n-4)afYax2+c dx, n> 1. (n - 1) c xn- 2 dx Yax2 + c 90. f xnYax2 + c = - c(n - 1)xn- 1

(n - 2) a dx f n> 1. (n-1) c xn- 2Yax2 + c'

91. f(ax2 + c)3/2 dx = ~ (2ax2 + 5c)Yax2 + c 3c2 + 8~ log(x~ + Yax2 + c), a> O.

a

xn+l(ax2 + c)3/2 3c f . 96. xn(ax2 + c) 3/2 dx = + -- xnYax2 + c dx. f n+4 n+4 376 • Appendix: Table of Integrals

xdx 1 97. f (ax2 + e)3/2 -

X2 dx x 98. f (ax2 + e)3/2 aYax2 + e 1 + ~ / log(x~+ Yax2 + e), a>O. ava 2 99 f x dx x • (ax2 + e)3/2 aYax2 + e 1 a + a~ Sin- ( XJ -e } a

x3 dx x2 2 100 = - + -Yax2 + e • f (ax2 + e)3/2 aYax2 + e a2 . n 101. f dx = l..-IOg( x ). x(axn + e) en axn + e n 102. f dx = ! f dx - !!:.. f x dx . (axn + e)m e (axn + e)m-l e (axn + e)m n ~) 103 f dx = _1_ 1 (Y ax + e - 0 • xYaxn + e n~ og Yaxn+ e+ ~' e>.

n dx - 2 -I ax 0 104· ~ /- ~ r-- sec J- ,e< . f xv axn + e n v -e e

105. f xm- I(axn + e)P dx

= 1 [xm(axn + e)P + npefxm-I(axn + e)P-1 dx] m+ np

= 1 [-xm(axn + e)p+1 + (nt + np + n) fxm-I(axn + e)p+l dx]. en(p + 1)

1 [xm-n(axn + e)P'+1 - (m - n)efxm-n-I(axn + e)P dx]. a(m + np)

= _1_[xm(axn + e)p+1 - (m + np + n) fxm+n-1(axn + e)P dx]. me

Expressions Containing ax2 + bx + C 2 108 f dx = 1 1 (2ax+b-Yb -4ae) . ax2 + bx + e Yb 2 - 4ae og 2ax + b + Yb 2 - 4ae ' • Appendix: Table ofIntegrals 377

109 f dx - 2 t -1 2ax + b b2 < 4aeo · ax2 + bx + e - V 4ae - b2 an V 4ae - b2'

110. f dx = _ 2 , b2 = 4aeo ax2 + bx + e 2 ax + b 111 f dx = 2ax + b · (ax2 + bx + e)n+l n(4ae - b2)(ax2 + bx + e)n 2(2n - l)a f dx + n(4ae - b2) (ax2 + bx + e)n°

xdx 1 2 b dx 112. = -log(ax + bx + e) -- f 0 fax2 + bx + e 2a 2a ax2 + bx + e x2 dx x b 113. --=----- = - - -log(ax2 + bx + e) fax2 + bx + e a 2a2 2 b - 2 ae f dx + 2 a2 ax2 + bx + eO xndx xn- 1 e f xn- 2 dx b f xn- 1 dx 114. fax2 + bx + e = (n - 1) a - -; ax2 + bx + e - -; ax2 + bx + eO 115 f xdx = -(2c + bx) · (ax2 + bx + e)n+l n(4 ae - b2) (ax2 + bx + e)n b(2n - 1) f dx n(4 ae - b2) (ax2 + bx + e)n° xm dx xm- 1 116 = ------· f (ax2 + bx + e)n+l a(2n - m + 1) (ax2 + bx + e)n m 1 n - m + 1 b f x - dx 2n- m+ 1 a (ax2 + bx+ e)n+l m + m - 1 e f x - 2 dx 2n - m + 1 a (ax2 + bx + e)n+l 2 117. dx = ~ 10 ( x )- ~ dx f 2 2 f 2 x(ax + bx + e) 2e g ax + bx + e 2e (ax + bx + e) 0 2 ~ 118 f dx = _b_ 10 (ax + bx + e) _ · x2(ax2 + bx + e) 2e2 g x2 ex b2 a) dx 2 ~ f 2 + ( 2e - (ax + bx + e) 0

119. f dx = _ 1 xm(ax2 + bx + e)n+l (m - l)exm - 1(ax2 + bx + e)n (n + m - 1) b f dx m - 1 e xm- 1 (ax2 + bx + e)n+l (2n + m - 1) a f dx m - 1 e xm- 2(ax2 + bx + e)n+l . 378 • Appendix: Table of Integrals

120. I dx = 1 x(ax2 + bx + c)n 2c(n - 1) (ax2 + bx + c)n-l bI dx II dx - 2c (ax2 + bx + c)n + -; x(ax2 + bx + c)n-l .

121. I 2 dx = ~ ~ log(2ax + b + 2~Yax2 + bx + C), a>O. ax + bx + c va

122 I dx - _1_ . -1 -2ax - b 0 • Yax2 + bx + c - ~ sm Yb2 _ 4ac' a < . 2 123 I x dx = Yax + bx + c - ~ I dx • Yax2 + bx + c a 2a Yax2 + bx + c· xn dx xn- 1 2 124. Y 2 = --Yax + bx + c I ax + bx + c an _ b(2n - 1) I xn- 1 dx _ c(n - 1) I xn- 2 dx 2an Yax2 + bx + c an Yax2 + bx + c· 2ax + b 125. Yax2 + bx + c dx = Yax2 + bx + c I 4a 4ac - b2 I dx + Sa Yax2 + bx + c·

(ax2 + bx + c) 3/2 b I 126. xYax2 + bx+ cdx= -- Yax2 + bx+ cdx. I 3a 2a 2 127. Ix2Yax2 + bx + c dx = (x _ !!!!..) (ax + bx + c)3/2 • 6a 4a (5b2 - 4ac) I' + 2 Yax2 + bx + c dx. I6a dx - 1 1 ax2 + bx + c + ~ 128. -- ~ r og(Y I xYax2 + bx + eve x

+ 2~). c> O.

129 I dx - 1 . -I bx + 2c 0 • xYax2 + bx+ c - ~ sm xYb2 _ 4ac' c<.

130 dx - -~Y 2 b • IxYax2 + bx - bx ax + x, c= O. 2 131. I ,dx = _ Yax + bx + c xnYax2 + bx + c c(n - l)xn- I + b(3 - 2n) I dx + a(2 - n) I dx 2c(n - 1) xn- 1Yax2 + bx + c c(n - 1) xn- 2Yax2 + bx + c·

132 I dx = - 2(2ax + b) b2 =1= 4ac. • (ax2 + bx + c)3/2 (b 2 - 4ac)Yax2 + bx + c' • Appendix: Table ofIntegrals 379

I 133 dx b2 = 4ae. · I (ax2 + bx + e)3/2

Miscellaneous

134. IV2px- x2 dx= ~[(x- P)V2px- x2 + p2 sin -I(X; P)J.

135. dx 2 = cos-I(~). Iv2px - x P

136 I dx = ·Vax + bVex + d 2 -IJ-e(aX + b) 2 h-IJ e(ax + b) ~ ,r-- tan or ~ r- tan v - ae a( ex + d) v ae a( ex + d) .

137. IVax+ bVex+ ddx= (2aex+ be+ ad)~~ 4ac (ad - be)2 I dx + 8ae Vax + bVex+ d' 138. I)ex+ d dx= Vax + b~ + (ad- be) I dx . ax + b a 2a Vax + b Vex + d

139. I) x + b dx = v;+d -v;+b + (b - d) log(v;+d + -v;+b). x+ d

140. I)I + x dx = sin-Ix - VI - x2. I- x

141. I)p - x dx = vp=-;e ~ + (P + q) sin-I) x + q. q+x p+q

142. I)P+ x dx= -v;;+;:~+(P+ q) sin-I)q- x. q-x p+q

143. V d~ = 2 sin x - p. I x-p q-x -1) q-p

Expressions Containing sin ax

144. Isin u du = -cos u, where u is any function of x.

145. I sin ax dx = - ~ cos ax.

· 2 d x sin 2ax 146· SIn ax x = - - . I 2 4a 380 • Appendix: Table of Integrals

147. ISin3 ax dx = _l cos ax + -.!..- cos3ax. a 3a 3 . 4 d 3x 3 sin 2ax sin axcos ax 148. SIll ax x = - - - . I 8 16a 4a

sinn-laxcos ax n - 1 n-2 d 149. Isinnax dx = ------+ --I'Sill ax x (n pos. integer). na n

150. -.--dx = -1log ( tan -ax) = -loge1 esc ax - ctn ax). ISIll ax a 2 a

151. .dx 2 = Icsc2ax dx = --1 ctn ax. ISill ax a dx 1 cos ax n - 2 I dx 152. ------+ -- n integer> 1. Isinnax a(n - 1) sinn-lax n - 1 sinn- 2ax'

153.

154.

155.

157. II + sin x dx = ~2(sin ;- cos ;} use + when

(8k - 1); < x$ (8k + 3) ;, otherwise -, k an integer.

158. IVI - sin xdx= ~2(sin; + cos ;); use + sign when

(8k - 3) 7T < x $ (8k + 1) 7T, otherwise -, k an integer. 2 2

Expressions Involving cos ax

159. Icos u du = sin u, where u is any function of x.

160. I cos ax dx = : sin ax.

x sin 2ax 161. cos2ax dx = - + . I 2 4a • Appendix: Table ofIntegrals 381

162. fcos3ax dx = .l sin ax - _1_ sin3ax. a 3a 3 4 d 3x 3 sin 2ax cos axsin ax 163. cos ax x = - + + . f 8 16a 4a cosn-laxsin ax n - 1 164. cosnax dx = + --fcosn-2ax dx (n pos. integer). f na n

165. _d_x_ = .llOg(tan(_a_x + 7T)) = .llog(tan ax + sec ax). f cos ax a 2 4 a

166. d: = .l tan ax. fcos ax a dx 1 sin ax n - 2 f dx 167. ------+ -- n integer> 1. f cosn ax a(n - 1) cosn-1ax n - 1 cosn-2ax' dx 1 ax 168. f 1 + cos ax = -;;, tan 2' dx 1 ax 169. f 1 _ cos ax = --;;, ctn 2'

170. f VI + cos x dx = ±v2 f cos ; dx = ±2v2 sin ;; use + when

(4k - 1) 7T < x::5 (4k + 1) 7T, otherwise -, k an integer.

171. f VI - cos x dx = ±v2 f sin ; dx = =+=2v2 cos ;; use top signs

when 4k7T < x::5 (4k + 2) 7T, otherwise bottom signs.

2 2 dx 1 1( Vb - c sin ax ) 172. f - tan- b + c cos ax aYb2 - c2 c + b cos ax ' 2 2 dx 1 1( Y c - b sin ax ) , 173. f - Y 2 tanh- b + c cos ax a c2 - b c + b cos ax sin(a - b) x sin(a + b) x 174. f cos ax cos bxd x = 2(a _ b) + 2(a + b) ,

Expressions Containing sin ax and cos ax . 1 (cos(a - b)x cos(a + b)X) 175. sm ax cos bx dx = -- + f 2 a-b a + b '

176. fsinnax cos ax dx = 1 sinn+ 1ax n=l=-1. a(n + 1) , 1 177. f cosnax sin ax dx = ----cosn+ 1ax, a(n+ 1) 382 • Appendix: Table of Integrals

sin ax 1 178. dx = --log(cos ax). Jcos ax a cos ax 1 . 179. . dx = -log(sm ax). J sm ax a

180. J(b+ csinax)ncosaxdx= 1 (b+ csinax)n+l, ni=-1. ac(n + 1)

181.J(b+ccosax)nsinaxdx= 1 (b+ccosax)n+1 ni=-1. ac(n + 1) ,

182. J cos ax dx = ~ Inlb + csin alx. b + csin ax ac sin ax 1 183. dx = --log(b + ccos ax). Jb + ccos ax ac

184. dx = 1 [log(tan !(ax + tan- I J bsin ax + ccos ax aYb2 + c2 2 '£)].b

dx -1. -I U. 185. = Mn Jb + ccos ax + dsin ax aYfi2 - (2 - d2 ' c2 + d2 + b(ccos ax + asin ax) 1 U= Y c2 + d2(b + ccos ax + dsin ax) or aYc2 + d2 _ b2 10g(V), c2 + d2 + b( c cos ax + d sin ax) + Y c2 + d2 - b2(c sin ax - d cos ax) V= Y c2 + d2(b + ccos ax + dsin ax)

b2 i= c2 + d2, - 7T < ax < 7T.

186. J dx = b + c cos ax + d sin ax

1 (b- (c+ d) cosax+ (c- d) sin ax) b2 = c2 + d2. ab b+ (c- d) cos ax+ (c+ d) sin ax '

187. sin2ax dx ---1 jH;+---tanC -l({;f--tan ax)--.X J b+ ccos2ax ac b b+ c c sin ax cos ax dx 1 188. 2 2 = 10g(bcos2ax+csin2ax). J bcos ax + csin ax 2a(c - b)

dx = _1_ 10g( bcos ax + csin ax). 189. J b2 cos2ax - c2 sin2ax 2abc b cos ax - csin ax

190. J dx , = _1_ tan- I ( c tan ax). b2 cos2ax - c2 sin2ax abc b . x sin 4ax 191. sm2ax cos2ax dx = - - . J 8 32a dx 1 192. = -log(tan ax). Jsin ax cos ax a dx 1 193. . 2 2 = -(tan ax - ctn ax). Jsm axcos ax a • Appendix: Table ofIntegrals 383

2 194. _si_n_a_x dx = .!.[-sin ax + IOg(tan(_a_x + 1T))]. J cos ax a 2 4 ax 195. c~s2ax dx = '!'[cos ax + IOg(tan-_ )]. J sm ax a 2 sinm-lax cosn+1ax 196. sinmaxcosnaxdx= -( J am + n)

+ m-l Jsinm-2axcosnaxdx, m, n>O. m+n sinm+laxcosn-lax 197. sinmaxcosnaxdx= ------• J a(m + n)

+ n - 1 Jsinmaxcosn-2ax dx, m, n>O. m+n

sinmax sinm+ 1ax 198. dx = I J cosnax a(n - l)cosn- ax m - n + 2 J sinmax 2 dx, m, n> 0, n *- 1. n - 1 cosn- ax cosnax _cosn+lax 199. dx= 1 J sinmax a(m - 1) sinm- ax m - n - 2 J cosnax + 2 dx, m, n, >0, m *- 1. (m - 1) sinm- ax 1 1 200. J dx = sinmax cosnax a(n- 1) sinm-lax cosn- 1ax

+ m+n-2J dx (n - 1) sinmaxcosn-2ax' 1 1 201. J dx = sinmaxcosnax a(m - 1) sinm-lax cosn- 1ax

m+n-2J dx + (m - 1) sinm-2axcosnax' 2n 2 Sin ax J (l - cos ax)n 202. dx = dx. (Expand, divide, and use 205). J cosax cos ax cos2nax J (l - sin2ax)n 203. . dx = . dx. (Expand, divide, and use 190). J sm ax sm ax sin2n+1ax cos2ax)n 204. dx = J(l - sin ax dx. (Expand, divide, and use 218). J cos ax cos ax cos2n+lax f(1 - sin2ax)n 205. f . dx = . cos ax dx. (Expand, divide, and use 217). sm ax smax 384 • Appendix: Table of Integrals

Expressions Containing tan ax or ctn ax (tan ax = lldn ax)

206. J tan u du = -log(cos u) or log(sec u), where u is any function of x.

207. Jtan ax dx = -: log(cos ax).

208. J tan2ax dx = : tan ax - x.

1 1 209. Jtan3ax dx = 2a tan2ax + -;;, log(cos ax).

210. Jtannax dx = 1 tann-1ax - Jtann- 2ax dx, n integer> 1. a(n - 1)

211. Jctnudu=log(sinu) or -log(cscu), where uis any function ofx.

212. Jctn2 ax dx = J d: = _..!.. ctn ax - x. tan ax a

213. Jctn3ax dx = __1_ ctn2ax - ..!..log(sin ax). 2a a

214. Jctnnax dx = dx Jtannax

- 1 ctnn-l ax - Jctnn- 2ax dx, n integer> 1. a(n - 1) dx J ctn ax dx 215. J b + c tan ax - b ctn ax + c

= b2 : c2 [bX + : log(bcos ax + csin ax) J dx J tan ax dx 216. J b + c ctn ax - b tan ax + c 1 = 2 2(bX-'£log(CCOsax+bsinaX)). b + c a

-d==;;==x 1. c. ) 217. ----;=== = SIn- I(H---Sill ax J Yb+ctan2ax aYb-c b ' • Appendix: Table ofIntegrals 385

Expressions Containing sec ax = 1/ cos ax or esc ax = l/sin ax

218. Isec u du = log(sec u + tan u) = IOg[ tan (; + :)l where u is any function of x. ~ ~x 219. Isec ax dx = log[tan( + :)1

220. I sec2ax dx = ~ tan ax.

I 3 ~ 221. I sec ax dx = 2a [tan ax sec ax + log ( tan( + :))1 1 sin ax 222. secnax dx = ------::• I a(n - 1) cosn-lax + n - 2 Isecn-2ax dx, n integer> 1. n-I

223. I csc u du = log(csc u - ctn u) = log(tan ;}

where u is any function of x.

224. I csc ax dx = ~ log(tan ~x)-

225. I csc2ax dx = - ~ ctn ax.

I 3 ~x) 226. I csc ax dx = 2a [- ctn ax csc ax + log(tan 1 1 cosax n- 2 I 227. cscnaxdx= + -- cscn- 2axdx n integer> 1. I a(n - 1) sinn-lax n - 1 '

Expressions Containing tan ax and sec ax or ctn ax and esc ax

228. I tan u sec u du = sec u, where u is any function of x.

229. I tan ax sec ax dx = ~ sec ax.

230. Itannaxsec2 ax dx = 1 tann+lax, a(n + 1) 386 • Appendix: Table of Integrals

231. Itan ax secnax dx = .l.... secnax, an

232. Ictn u csc u du = -csc u, where u is any function of x.

233. I cm ax csc ax dx = - ~ csc ax.

1 234. I ctnnax csc2ax dx = ------ctnn+la~ n=l=-l. a(n + 1)

235. Ictn ax cscnax dx = _.l.... cscnax, an csc2ax dx 1 236. = --}og(ctn ax). I ctn ax a

Expressions Containing Algebraic and

d 1. 1 237• xsm· ax x = 2sm ax - -xcos ax. I a a 2 2' d 2x . 2 x 238· x sm ax x = 2 sm ax + "3 cos ax -- cos ax. I a a a 2 3 3' d 3x . 6. x 6x 239· x SIn ax x = --2- sm ax - 4 sm ax -- cos ax + "3 cos ax. I a a a a . 2 d x2 xsin 2ax cos 2ax 240. I x sm ax x = 4 - 4a - 8a2 .

241. x2 sin2ax dx = ~ - (~ - __1_) sin 2ax _ x cos 2ax I 6 4a 8a3 4a2 4 2 242. x3sin2 ax dx = x - (~- 3X) sin 2ax _ (3X - cos 2ax. I 8 4a 8a3 8a2 _3_)16a4

. 3 d xcos 3ax sin 3ax 3xcos ax 3 sin ax 243. x SIn ax x = - - + . I 12a 36a2 4a 4a2

244. I xnsin ax dx = - ~ xn cos ax + : I xn- l cos ax dx, n> O.

245. Isin ax dx = ax _ (ax)3 + (ax)5 _ .... x 3·31 5·51

246. I sin ax dx = - 1 sin ~x + a I cos aX dx . xm (m - 1) xm - (m - 1) xm - l

247. Ix cos ax dx = ~ cos ax + .l x sin ax. a a • Appendix: Table ofIntegrals 387

2 2 d 2x 2. x . 248•X cos ax x = 2 cos ax - 3 sm ax + - sm ax. f a a a 2 3 3 (3a2x2 - 6) cos ax (a x - 6x) sin a 249. x cos ax dx = 4 + 3 . f a a 2 250 f 2 d x x sin 2ax cos 2ax • x cos ax x = 4 + 4a + 8a2 . 2 251. x2 cos2axdx=-+x3 (----x 1).sm2ax+ x cos 2ax. f 6 4a 8a3 4a2 4 2 252. x3 cos2ax dx = x + (~- 3X) sin 2ax + (3X - _3_) cos 2ax. f 8 4a 8a3 8a2 16a4

3 d - xsin 3ax cos 3ax 3xsin ax 3 cos ax 253. f xcos ax x- 12a + 36a2 + 4a + 4a2 .

254. fxn cos ax dx = .lxn sin ax - !!:... fxn- 1 sin ax dx, n pos. a .! a

255. fCOS ax dx = log(ax) _ (ax)2 + (ax)4 _ .... x 2·21 4'4!

256. f cos ax dx = 1 cos ax a fsin ax dx x m (m - 1) xm- 1 (m - 1) x m- 1 '

Expressions Containing Exponential and Logarithmic Functions

U 257. feu du = e , where u is any function of x.

bU 258. bU du = , where u is any function of x. f log(b) bax ax ~ ax ax 259. f e dx = e , f b dx = a log(b) . xbax 260. f xe ax dx = ~; (ax - 1), xbaxdx=--• f a log(b)

263.

n pos. integ. 388 • Appendix: Table of Integrals

ax 264. fxne-axdx= - e- [(ax)n + n(ax)n-l + n(n-l)(ax)n-2 an+ 1 + ... + n!J, n pos. integ.

xnbax n f 265. xnbaxdx= - xn-1baxdx, npos. f a log(b) a log(b) eax (ax)2 (ax)3 266. - dx = log(x) + ax + -- + --+ .... f x 2· 2! 3· 3! ax eax 1 (e f eax ) 267. -dx=-- --- + a --dx n integ. > 1. f xn n - 1 xn- 1 xn- 1 ' dx 1 268. = -[ax - log(b + eeax)]. fb + ee ax ab eax dx 1 269. = -log(b + ee ax). f b + ee ax ae

270. dx = --1 tan-1(ft)eax -, band e pos. fbe ax + ce-ax a~ e

271. f eax sin bx dx = a2e;x b2 (a sin bx - b cos bx).

ax . b' d - eaX[(b- e) sin(b- e)x+ acos(b- e)x] 272. f e sm x sm ex x - 2 [a2 + (b _ e) 2] eax[(b+ e) sin(b+ e) + acos(b+ e)x] 2[a2 + (b + e)2]

273. f eax cos bx dx = a2e;x b2(a cos bx + bsin bx).

274 fax b d - eaX[(b - e) sin(b - e)x + acos(b - e)x] • e cos xcos ex x - 2[a2 + (b _ e)2] eaX[(b + e) sin(b + e)x + acos(b + e)x] + -..::.....:....----'-----'------'------'-----'---=- 2[a2 + (b + e)2] 275 fax . b d - eaX[a sin(b - e)x - (b - e) cos(b - e)x] • e sm xcos ex x - 2[a2 + (b _ e)2] eaX[a sin(b + e)x - (b + e) cos(b + e)x] + 2[a2 + (b + e)2] eax cos e 276. eax sin bx sin(bx + e) dx = --• f 2a eax[acos(2bx + e) + 2bsin(2bx + e)] 2 2(a + 4b2 ) eax cos e 277. eax cos bx cos(bx + e) dx = --• f 2a eax[acos(2bx+ e) + 2bsin(2bx+ e)] +--"----'------'------'------'-=-- 2(a2 + 4b2 ) • Appendix: Table ofIntegrals 389

eax sin c 278. f eax sin bx cos(bx + c) dx = 2a + eax[a sin(2bx + c) - 2b cos(2bx + c)] 2(a2 + 4b2 ) eax sin c 279. eax cos bx sin(bx + c) dx = --• f 2a eax[a sin(2bx + c) - 2b cos(2bx + c)] +---'-----'-----'-----::----'----'---'- 2(a2 + 4b2 ) xeax 280. xeax sin bx dx = 2 2 (a sin bx - b cos bx) f a + b eax 2 2 2 [(a2 - b2) sin bx - 2 ab cos bx]. (a + b ) xeax ax 281. f xe cos bx dx = 2 2 (a cos bx + b sin bx) a + b eax 2 2 2 [(a2 - b2) cos bx + 2 absin bx]. (a + b ) ax n 1 e (cos - bx) (a cos bx + nb sin bx) a2 + n2b2 n(n-1)b2 f + eax cosn-2bx dx. a2 + n2b2 . b d eaX(sinn-1bx)(asinbx- nbcosbx) 283• eax SInn XX = ---'------'---:------c::--:::------'- f a2 + n2b2

284. flog(ax) dx = x log(ax) - x.

x2 x2 285. f x log(ax) dx = 2 log(ax) - 4·

x3 x 3 286. fx2 1og(ax) dx = 31og(ax) - gO

287. f[log(ax)]2 dx= x[log(ax)]2 - 2xlog(ax) + 2xo

288. f [log(ax)] n dx = x[log(ax)] n - n f [log(ax)] n-l dx, n pos.

n n 1 lOg(aX) 1) 289. x log(ax) dx = x + - , n -=1= -1. f ( n + 1 (n + 1)2 xn+l m f 290. xn[log(ax)]m dx = --[log(ax)]m - -- xn[log(ax)]m-l dx. f n + 1 n+ 1 390 • Appendix: Table of Integrals

[log(ax)]n [log(ax)]n+l 291. x dx = - n-=l= -1. f n+I

292. f dx = log[log(ax)]. x log(ax)

293. f dx = _ 1 . x(log(ax»n (n - 1) (log(ax»n-l n 294. f x dx = (log(ax) )m -xn+l n + 1 f xndx (m-I)(log(ax»m-l + m-I (log(ax»m-l' m-=l=1.

n Y 295 f x dx = _1_ f e dy y = (n + 1) Inlaxl. • log(ax) an+l y'

xndx 1 296. 1 ( = --.;}[loglog(ax)II+ (n + 1) log(ax) f og ax) an + (n + I)2[log(ax)]2 + (n + I)3[log(ax)]3 + ... ]. 2· 2! 3· 3!

297. dx = flog(ax) ![log[log(ax)] + log(ax) + (log(ax»2 + (log(ax»3 + ''']. a 2·21 3· 3!

298. fsin[log(ax)] dx = ; [sin[log(ax)] - cos[log(ax)].

299. fcos[log(ax)] dx= ;[sin[log(ax)] + cos[log(ax)].

ax 300. eax log(bx) dx = -e1 ax log(bx) ---Ife dx. f a a x

Expressions Containing Inverse Trigonometric Functions

301. fsin-Iaxdx= xsin-Iax+ :VI - a2x 2.

302. f (sin- Iax)2 dx = x(sin- Iax)2 - 2x + ~VI - a2x 2 sin-lax.

303. fxsin-Iax dx = ~ sin-lax - _1_ sin-lax + ~VI - a2x2. 2 4a2 4a xn+l a f xn+l dx 304. xn sin-lax dx = --sin-lax - -- , n -=1= -1. f n + 1 n + 1 VI - a2x2 • Appendix: Table ofIntegrals 391

sin-Iax dx 1 ()3 1 ·3 ( )5 305· f = ax+ ax + ax x 2·3·3 2·4·5·5 1·3·5 + (ax)7+... a2x2 <1. 2·4·6·7·7 ' 2 2 sin-Iax dx 1. -1 I (1 + VI - a x ) 306• = -- SIn ax - a og . f x2 x ax

307. fcos-Iaxdx= xcos-Iax- ~Vl - a2x2.

308. f (cos-Iax)2 dx = x(cos-Iax)2 - 2x - ~Vl - a2x2 cos-lax.

x2 1 x 309. xcos-Iax dx = - cos-lax -- cos-lax - -VI - a2x2. f 2 4a2 4a xn+l a f xn+l dx 310. xn cos-lax dx = --cos-lax + -- , f n + 1 n + 1 VI - a2x2 n * -1.

cos-Iax dx 7T II 1 311. f = -In ax - ax - (ax)3 x 2 2·3·3

1·3 1·3·5 7 2·4·5· 5 (ax)5 - 2.4.6.7.7 (ax) - .'., 2 2 cos-Iax dx __1- -1 I (1 + VI - a x ) 312• 2 - cos ax + a og . f x x a

313. ftan-Iax dx = x tan-lax - -.L log(l + a2x2). 2a

n * -1.

n l n xn+l a f x + dx 317. x ctn-Iax dx = --ctn-Iax + -- , n -1. f n + 1 n + 1 1 + a2x2 * 2 2 ctn-Iax dx __1- -1 !!:...I (1 + a x ) 318•f 2 - ctn ax + og 2 2 . X x 2 a x

319. fsec-Iaxdx= xsec-Iax- ~ log(ax+ Va2x2 -1).

xn+ 1 1 f xn dx 320. xn sec-lax dx = --sec-lax ± -- , n* -1; f n + 1 n + 1 Va2x2 - 1 use + sign when 7T/2 < sec-l ax < 7T; - sign when 0 < sec-lax < 7T/2. 392 • Appendix: Table of Integrals

321. fese-laxdx= xese-lax+ ~ log(ax+ Ya2x2 -1).

n xn+l 1 f xndx 322. f x esc-lax dx = --esc-lax ± -- ,n=1= -1; n + 1 n + 1 Ya2x 2 - 1 use + sign when 0 < ese-l ax< 1T/2; - sign when -1T/2 < esc-lax < O. ----1.1--• Index

Absolute extrema, 107-110 graphs of, 191-192 equating, 218-225 Accumulated , 148 for, 179 finding, 39-50 Accumulation functions relationships of, 179-181 rectangular approach to, defined by f, 58, 149 Scalar Multiple Rule for, 187 39-50, 43-45 using partial Riemann simple, using Trapezoidal representing between two sums to approximate Rule on functions by a definite in• accumulation func• without, 274-276 tegral,167-169 tions, 163-167 specific, 190-192 of , 56 , 216 substitution for finding, , horizontal, 304 C in, 181 224-225 Average of change, 31, 67 calculating, 178-192 Sum Rule for, 187-188 closed forms for, 331 and the , Base a, 125-126 defining, 147 274-276 , 92 finding using CAS for comparing, of basic functions, 195 C, in antiderivatives, 181 181-186 Antidifferentiation, 178 , Fundamental of general exponential Arccos, 303 Theorem of. See functions, 194-195 Arcsin, 302 Fundamental Theorem of linear combinations, Arctan, 304 of Calculus 187-190 formulas, 150 CAS (computer system) of trigonometric func• Area function, cumulative, 58 calculating antiderivatives tions, 193, 195, with, 195 243-244 approximating, under a finding with, general, 181 , 148 142-143 393 394 • Index

Chain Rule, 86-88, 89, 237, of a function, inspecting the domain of, 248, 309, 318 149-150 (See also 17-22 , 217 Antiderivatives) and integrals, 3-64 Circle, of, 267 applying a rectangular ap• ofinverse functions, 305-309 Closed forms, for antideriva- proach to, 43-45 left-hand, 18 tives, 331 approximating, 271-292 of local extema, 107 Combinations and calculating Riemann of logarithmic functions, differentiating, 68-69 sums, 50-54 135-139 linear, finding antideriva• between deriva• for, tives of, 181-186 tives and, 59-61 34-36 Composition of functions, 68 defining, 49, 148 of natural logs, 126, sytem integrating functions 131-134 (CAS). See CAS whose graphs dip be• negative, 100 Concavity, of functions, 93-98 low the axis, 156-159 for, 69-70, Constant functions, 72 interpreting, 54-58, 115-116 Constant Rule, 68-69, 88 160-163 nth,97 Continuous functions, 276 representing the area be• of powers of e, 126-130 Converging integrals, 346 tween two curves by, proportional, 340 Cosecant functions 167-178 relationship between func- antiderivatives of, 245 approach, tions and, 22-25 derivatives of, 140-141 148-149, 150-156 representing by expres- Cosine functions using partial Riemann sions, 12-17 antiderivatives of, 245-246 sums to approximate right-hand, 18 derivatives of, 135 accumulation func• second,98 inverse, 302 tions, 163-167 signs and sign charts, 24, Cotangent functions Degree of ordinary differen• 26-27, 82, 100 antiderivatives of, 245-246 tial , 322 of sin (x) and cos(x), derivatives of, 140-141 x, 48 119-122 Critical off, 25 Derivatives, 66-68 and, 5-8 Cumulative area function, 58 calculus approach to, third,98 Curves 68-145 zeros of, 82 families of, 326 connection between defi• Difference quotients, 12 fitting to a data set, 217, nite integrals and, Diff~rence Rule, 75-77,89 282-284 59-61 Differences of functions, 68 representing areas be• definition of, 9-12 Differentiable functions, 12, tween, by definite inte• of derivatives, 98 309 grals, 167-169 domains of, 17-22 Differential equations, 296 and the Riemann sum ap• equations involving, investigating, 323-325 proach to approximate 317-337 ordinary (ODE), 322 the area under a of exponential functions, solving, using separation of curve, 155 125-130, 139 variables, 326-331 sketching, 101-107 of functions containing Differentials, 225 Cylinders, right circular, 352 trigonometric expres• Differentiation Cylindrically shaped solids, sions, 122-125, 139 of combinations of func• 361-362 general, 12 tions, 68-89 of general exponential and implicit (See Implicit differ• Data sets, using the logarithmic functions, entiation) Trapezoidal Rule on, 135-139 integration and, 189 276-280 graphs of functions and, of inverse trigonometric Decay constant, 289 26-27 functions, 305-309 Decreasing functions, 22, 99 higher~rder, 98-101 logarithmic, 316 Definite integrals, 38-61, increasing/decreasing, reversing the process of, 149 147-213 97-98 rules of, 88-89, 139 • Index 395

Diminishing returns, point Extended Rule, 78-80, instantaneous rate of of,36 89, 236 change, 32, 67 Disc approach, to find the Extrema integrable, 54 of solids of absolute, 107-110 inverse (See Inverse func• revolution, 350-359 local, 107 tions) Discontinuity, jump, 19 logarithmic (See Discrete functions, 276 Families of curves, 326 Logarithmic func• Distance Filling a Nectar Bottle, pro• tions) finding, 151 ject, 362-365 power, 72 formula for, 150 Finding an Average products of, 68 varying and, Temperature, project, proportional, 340 151-156 285-292 quotients of, 68 Distance-to-date, 152 Finding the Right Water rate of change of, 113 Diverging integrals, 346 Level, project, 266-269 relationship between deriv- DNE (term), 23 Finite value, 344 atives and, 22-25 Domains, of derivatives, First Test, The, scalar multiples of, 72 17-22 25-26, 91 secant, 140, 141 Functional behavior, analyz• sign charts for, 82 e, 126-130 ing, 93-119 (See Sine functions) End-points of subintervals, Functions sums of, 67 46-47 accumulation (See (See Tangent func• Equating areas, 218-225 Accumulation func• tions) Equations tions) trigonometric (See differential, 296, 323-331 average rate of change Trigonometric func• involving derivatives, and, 31, 67 tions) 317-337 basic, finding antideriva- velocity, 151-156 Equivalent integrals, 217, 229 tives of, 181-186 whose graphs dip below Estimating the National Debt, combinations of, 67-68 axis, integrating, project, 280-281 composition of, 68 156-160 's Formula, 336 concavity of, 93-98 without simple antideriva• Euler's method, utilizing, constant, 72 tives, using 331-337 contintlOus, 276 Trapezoidal Rule on, Explicit solutions, 325 cosecant (See Cosecant 271-274 Exponential decay function, functions) Fundamental Theorem of 289 cosine (See Cosine func• Calculus (FTC), 58, Exponential form of loga• tions) 61, 150, 196-197, 209, rithms, 125 cotangent (See Cotangent 216 Exponential functions, 67, functions) comparing accumulation 125 decreasing, 22, 99 functions and anti• antiderivatives of, 194-195, derivatives of (See derivatives with calcu• 244-245 Derivatives) lators, 197-204 derivatives of, 125-130, 139 differences of, 67 Part 1, 197-201,209 Expressions differentiable, 12, 309 Part II, 203-209 integrating, containing discrete, 276 trigonometric func• exponential (See General antiderivative off, tions, 195, 245 Exponential functions) 181 representing derivatives by, finding antiderivatives of, General derivatives of func• 12-17 181-186 tions, 12 in Tables of Integrals, graphs of, derivatives and, General exponential func• 369-392 22-27 tions trigonometric (See implicit, 297, 313 antiderivatives of, 194-195, Trigonometric expres• increasing, 22, 99 244 sions) inflection points of, 96-97 derivatives of, 135-137 396 • Index

General logarithmic func• tables, 263-264 project for, 256-260 tions, 137-139 expressions containing strategy underlying, Graphs algebraic and trigono• 250-253 of antiderivatives, 191 metric functions, using, 253-256 dipping below axis, inte• 386-387 by substitution, 217-243 grating functions of, expressions containing project for, 241-243 156-160 ax + b, 369-373 situations where substitu• of functions, derivatives expressions containing tion applies, 231-237 and,26-27 ax2 + bx + c, 376-379 strategy underlying, reflecting, 122, 299-304 expressions containing 225-231 ax2 + c, axn + c, x2 :t: using, 238-241 Higher-order derivatives, p2, and p2 - x2, summary of rules of, 98-101 373-376 260-261 Horizontal asymptotes, 304 expressions containing Inverse functions, 309-313 Horizontal Line Test, 299 exponentialandlog~ cosine, 302 Horizontal , 22 rithmic functions, finding the derivatives of, 387-390 309-313 i, as index of , 37 expressions containing sine, 301 Implicit differentiation, inverse trigonometric tangent, 304 297-298 functions, 390-392 trigonometric, 300-309 and defining inverse expressions containing IVP. See Initial Value Problem trigonometric func• sec ax =l/cos ax or tions, 300-304 csc ax =l/sin ax, 385 Jump discontinuity, 19 and differentiating inverse expressions containing trigonometric func• sin ax, 379-380 k, fixed constant, 187 tions, 305-309 expressions containing and finding the derivative tan ax and sec ax or Left-hand derivatives, 18 of an , ctn ax and csc ax, Leibnitz, Gottfried, 210 309-313 385-386 Linear combinations, finding using, 298-300 expressions containing antiderivatives of, Implicit functions, 297, 313 tan ax or ctn ax (tan 187-190 Implicit solutions, 325 ax = IIctn ax), 384 Local extrema, 107 Improper integrals, 296, expressions involving cos Logarithmic differentiation, 344-350 ax, 380-381 316 Increasing functions, 22, 99 expressions involving sin Logarithmic functions Indefinite integrals, 181 ax and cos ax, 381-383 base a, 125-126 Infinite extent, 344, 345 looking up integrals in, derivatives of, 131-134 Inflection points, 96-97 264-266 exponential form, 125 Initial value problem (IVP), miscellaneous algebraic general 330-331 expressions, 379 antiderivatives of, Instantaneous rate of change, project for, 266-269 135-137 32, 67 Integrands, 237 derivatives of, 194-195, Integrable functions, 54 Integration, 344 244 Integrals, 216 approximating definite in- converging, 346 tegrals (See Definite in• natural, 126 definite (See Definite inte- tegrals) properties, 304 grals) differentiation and, 186, diverging, 346 187 formula, 171 equivalent, 217, 229 evaluating improper inte- . See Extrema improper, 296, 344-350 grals, 345-350 Mean Value Theorem for indefinite, 181 methods of, 215-293 Derivatives, 34-35 Mean Value Theorem for, numerical, 216 Mean Value Theorem for 212 by parts Integrals, 212 • Index 397

Midpoint approach, 61 Reflections, of exponential finding using the washer Minima. See Extrema functions, 125 approach, 359-362 Modeling Template, 353 , 318 project for finding, Related rates problems, 296, 362-365 Natural logs, 126 319-322 Solutions, implicit/explicit, Negative derivatives, 100 Revolution, solids of. See 325 , Sir Isaac, 210 Solids of revolution Sounding Off, project, Newton's Method, 342-344 Riemann sum approach, 256-260 , 273 148-149, 150-156 Step size, 331 nth derivatives, 97 Riemann sums, 48 Subintervals, 43, 46-48, 216 , 216 calculating, 50-54 Substitution. See Integration, partial (See Partial Riemann by substitution sums) Sum Rule, 75-76, 89 Ordinary differential equa• tion (ODE). See Right circular cylinders, 352 Sum Rule for Antiderivatives, 187-188 Differential equations, Right-hand derivatives, 18 Sums ordinary of functions, 67 Scalar Multiple Rule, 73-74, Riemann (See Riemann C, 326 89 sums) Partial Riemann sums, 58, for Antiderivatives, 187 149, 163-167 with , 73-74 Point of diminishing returns, Scalar multiples of functions, Tables, of integrals. See 36 67 Integrals, tables Positive, derivatives, 100 Scatterplots, 276 Tangent functions, 303 Power functions, 72 Secant functions antiderivative, 246-247 Power Rule, 69-72, 77, 88 antiderivatives of, 246, derivatives, 140-141 Probability density function, 247 inverse, 304 144-145, 273 derivatives of, 140, 141 Tangent lines, 94 , 80-83, 89, 248 Secant lines, 6, 66 to curves, 5 Products, of functions, 68 Second derivatives, 94, 98 slopes of, 66 Projects Second , Third derivatives, 98 Estimating the National 116-117 intervals Debt, 280-281 Separable ordinary differen- dividing, 150-151 Filling a Nectar Bottle, tial equations, 326-331 Time step, 331 362-365 , 8 TL (term), 22 Finding an Average Sigma notation, 37 Tracking the Human Race, Temperature, 285-292 Sign charts project, 241-243 Finding the Right Water for derivatives, 26-27 Trapezoidal Rule Level, 266-269 for functions, 82 finding a formula for, Sounding Off, 256-260 Signs of derivatives, 26-27, 82 271-274 Tracking the Human Race, Sine functions project with, 280-281 241-243 antiderivatives, 185, using Proportional functions and 245-246 on data sets, 276-282 derivatives, 340 derivatives of, 119-122 on functions without inverse, 302 simple antiderivatives, , 83-86, 89 Sketching curves, 101-107 274-276 Quotients Slopes Trapezoids, 177 difference, 12, 66 derivatives and, 5 areas of, 77 of functions, 68 of the tangent line, 66, 94 and the trapezoidal ap• Solids of revolution, volumes proach, 62 Rate of change, 296 of,296-297 Trigonometric functions Rectangular approach, to finding using the disc ap• antiderivatives of functions finding areas, 43-45 proach, 350-359 containing, 246-247 398 • Index

Trigonomeuic functions (cont.) reversing rules for differen• Washer approach, to find the defining inverse, 300-304 tiating, 244-245 of solids of derivatives of functions con• revolution, 359-362 mining, 122-125, 139 Variables, change of, 217 Washers, 359 differentiating inverse, Velocity, 151-156 formulas, 151, 170-171 305-309 Volumes of solids of revolu• integrating expressions tion. See Solids of revo• conmining, 193, 247 lution Zeros of derivatives, 82