thomas 8 a

CHAPTER 8 TECHNIQUES OF INTEGRATION

8.1 BASIC INTEGRATION FORMULAS

1. ; 2 u C 2 8x 1 Cu 8x 1du 16x dx' '16x dx du

8x 1 u ##

#

2. ; 2 u C 2 1 3 sin x Cu 1 3 sin xdu 3 cos x dx' '3 cos x dx du

1 3 sin x u

3. 3 sin v cos v dv; 3 u du 3 u C 2(sin v) Cu sin vdu cos v dv' ' 23 $# $#

4. cot y csc y dy; u ( du) C Cu cot ydu csc y dy' '$ # $

# u4 4cot y% %

5. ; ln u ln 10 ln 2 ln 5u 8x 2du 16x dx

x 0 u 2, x 1 u 10' '

0 2

1 1016x dx du8x 2 u#

#

"!#

c dk k

6. ; du ln u ln 3 ln 1 ln 3u tan z

du sec z dzz u 1, z u 3

' '1

1

4 1

3 3 31

sec z dztan z u

4 3

#

c dk k

# "

1 1

7. ; 2 ln u C 2 ln x 1 Cu x

du dx2 du

' 'dx 2 dux x 1 x

dxx

u

"#

k k

"

8. ; 2 ln u C 2 ln x 1 Cu x 1du dx2 du

' ' 'dx dx 2 dux x x x 1 x

dxx

u

"#

k k

9. cot (3 7x) dx; cot u du ln sin u C ln sin (3 7x) Cu 3 7xdu 7 dx' ' k k k k" " "7 7 7

10. csc ( x 1) dx; csc u ln csc u cot u Cu x 1du dx' '11

1

k k du1 1"

ln csc ( x 1) cot ( x 1) C "1

k k1 111. e csc e 1 d ; csc u du ln csc u cot u C ln csc e 1 cot e 1 Cu e 1du e d

' ') ) ) ) k k ) ))

)

12. dx; cot u du ln sin u C ln sin (3 ln x) Cu 3 ln xdu' 'cot (3 ln x)

x dxx

k k k k

494 Chapter 8 Techniques of Integration

13. sec dt; 3 sec u du 3 ln sec u tan u C 3 ln sec tan Cu

du' 't 3 t t

3 3 3

t

dt3

k k

14. x sec x 5 dx; sec u du ln sec u tan u Cu x 5du 2x dx' 'a b k k # # " "# #

ln sec x 5 tan x 5 C "## #k ka b a b

15. csc (s ) ds; csc u du ln csc u cot u C ln csc (s ) cot (s ) Cu sdu ds' '

1 1 11 k k k k

16. csc d ; csc u du ln csc u cot u C ln csc cot Cu

du' '" " " "

"

) ) ) ))

)

)

#

#

) k k d

17. 2xe dx; e du e e e 2 1 1u x

du 2x dxx 0 u 0, x ln 2 u ln 2

' '0 0

ln 2 ln 2u u ln 2ln 2

0

x# c d

#

!

18. sin (y) e dy; e du e du e 1 eu cos y

du sin y dyy u 0, y u 1

' ' '1

1

2 0 1

1 0ucos y u u e

e

c d

1

#

!"

" "

1

19. e sec v dv; e du e C e Cu tan vdu sec v dv' 'tan v u u tan v#

#

20. ; 2e du 2e C 2e Cu t

du' 'e dt

t dt2 t

u u tt

21. 3 dx; 3 du 3 C Cu x 1du dx' 'x 1 u u

ln 3 ln 33 " x 1

22. dx; 2 du C Cu ln xdu

' '2 2 2x ln ln dx

x

uln x u ln x # #

23. ; 2 du C Cu w

du' '2 dw 2 2

w dww

ln ln w u w

##

# #

u

24. 10 d ; 10 du C Cu 2du 2 d' '2 u 10 10

ln 10 ln 10))

)

) " "# # #u 2)

25. ; 3 tan x C 3 tan 3u Cx 3udx 3 du' '9 du 3 dx

1 9u 1 x " "# #

26. ; 2 tan u C 2 tan (2x 1) Cu 2x 1du 2 dx' '4 dx 2 du

1 (2x 1) 1 u " "# #

27. ; sin u 0u 3x

du 3 dxx 0 u 0, x u

' '0 0

1 6 1 2 dx du1 9x 1 u

63 3 3 6 18 " "

#

" " "" "#

!# #

1 1

Section 8.1 Basic Integration Formulas 495

28. sin sin 0'0

1 dt t4 t 6

" "# #

"

!"#

1

29. ; sin u C sin s Cu sdu 2s ds' '2s ds du

1 s 1 u #

" " #% #

30. ; sin u C sin (2 ln x) Cu 2 ln xdu' '2 dx du

x 1 4 ln x 1 u2 dxx

" "# #

31. 5 sec 5x C 6 sec 5x C' '6 dx 6 dx 6x 25x 1 5x x 5 # # "#

" " 5

k k k k

32. sec C' dr rr r 9 3 3 #

" "

33. ; tan u C tan e Cu edu e dx' ' 'dx e dx du

e e e 1 u 1x x 2xx

" " #

x

xx

34. ; sec u C sec e Cu edu e dy' ' 'dy e dy

e 1 u u 1e e 1du a b2y

y

y y

" " # # k k

y

yy

35. ; sec u du ln sec u tan uu ln xdu

x 1 u 0, x e u' ' '

1dx du

x cos (ln x) cos udxx

3

e 3 3

30 0

13 c dk k

1

1 1

1

1$!

ln sec tan ln sec 0 tan 0 ln 2 3 ln (1) ln 2 3 k k 1 13 336. ; ln 1 4u C ln 1 4 ln x Cu ln xdu ln x dx

' ' 'ln x dx ln x dx dux 4x ln x x 1 4 ln x 1 4u 8 82

x #

#" " " ## #

a b k k a b

37. 8 ; 8 8 tan uu x 1du dx

x 1 u 0, x 2 u' ' '

1 18 dx dx du

x 2x 2 1 (x 1) 1 u2 2 1

0# # # " "

!

"

c d

8 tan 1 tan 0 8 0 2 a b " " 14 1

38. 2 ; 2 2 tan uu x 3du dx

x 2 u 1, x 4 u 1' ' '

2 2 1

4 4 12 dx dx du

x 6x 10 (x 3) 1 u 1# # # " "

"

c d

2 tan 1 tan ( 1) 2 c d " " 1 14 4 139. ; sin u C sin (t 2) Cu t 2du dt' ' '

dt dt dut 4t 3 1 (t 2) 1 u

" "#

#

#

40. ; sin u C sin ( 1) Cu 1du d' ' 'd d du

2 1 u1 ( 1)) )

) ) ) " "# #

#

) ) )

41. ; sec u C sec x 1 C,u x 1du dx' ' 'dx dx du

(x 1) x 2x (x 1) (x 1) 1 u u 1 " " # ##

k k k k

u x 1 1k k k k


42. ; sec u Cu x 2du dx' ' 'dx dx du

(x 2) x 4x 3 (x 2) (x 2) 1 u u 1 " # ##

k k

sec x 2 C, u x 2 1 " k k k k k k43. (sec x cot x) dx sec x 2 sec x cot x cot x dx sec x dx 2 csc x dx csc x 1 dx' ' ' # # # # #a b a b' ' tan x 2 ln csc x cot x cot x x C k k44. (csc x tan x) dx csc x 2 csc x tan x tan x dx csc x dx 2 sec x dx sec x 1 dx' ' ' ' ' # # # # #a b a b cot x 2 ln sec x tan x tan x x C k k45. csc x sin 3x dx (csc x)(sin 2x cos x sin x cos 2x) dx (csc x) 2 sin x cos x sin x cos 2x dx' ' ' a b# 2 cos x cos 2x dx [(1 cos 2x) cos 2x] dx (1 2 cos 2x) dx x sin 2x C ' ' 'a b#46. (sin 3x cos 2x cos 3x sin 2x) dx sin (3x 2x) dx sin x dx cos x C' ' '

47. dx 1 dx x ln x 1 C' 'xx 1 x 1

" k k48. dx 1 dx x tan x C' 'x

x 1 x 1#

# # " "

49. dx 2x dx x ln x 1 (9 ln 8) (2 ln 1) 7 ln 8' '

2 2

3 3 32

2x 2xx 1 x 1$

# # # # c dk k

50. dx (2x 3) dx x 3x ln 2x 3 (9 9 ln 9) (1 3 ln 1) ln 9 4' ' 1 1

3 34x 7 22x 3 2x 3#

# $" c dk k

51. dt (4t 1) dt 2t t 2 tan C' '4t t 16t 4 tt 4 t 4$ #

# #

#

# " 52. d 1 d ln 2 5 C' '2 7 7 5 52 5 2 5 3) ) ) ) )) )$ # $ #

# ##

) ) ) ) ) ) a b k k53. dx sin x 1 x C' ' '1 x dx x dx

1 x 1 x 1 x

" # # # #

54. dx (x 1) ln x C' ' 'x 2 x 12x x 1 2 x 1

dx dxx

"# k k

55. dx sec x sec x tan x dx tan x sec x 1 2 ( 1) 2' '0 0

4 41 1 1 sin x

cos x # %

!# ! a b c d 1

56. dx dx tan (2x) ln 1 4x' '0 0

1 2 1 2 2 8x 2 8x1 4x 1 4x 1 4x

" # "#!# # # c dk k

tan 1 ln 2 tan 0 ln 1 ln 2 a b a b" " 1457. dx dx sec x sec x tan x dx tan x sec x C' ' ' 'dx1 sin x 1 sin x cos x

(1 sin x) (1 sin x)

# a b# # a b58. 1 cos x 1 cos 2 2 cos sec dx tan C x x dx dx x x1 cos x 2 cos# # # # ## #"' ' '##

x

59. d d ; ln u C ln 1 sin Cu 1 sin du cos d' ' '"

sec tan udu

) )) ) )

)

) )

k k k k


60. d d ; ln u C ln 1 cos Cu 1 cos du sin d' ' '"

csc cot 1 cos usin du

) ) )

)) ) )

)

) )

k k k k

61. dx dx 1 dx 1 dx 1 csc x dx' ' ' ' '" " #1 sec x cos x 1 cos x 1 sin x sin xcos x 1 cos x cos x # # 1 csc x csc x cot x dx x cot x csc x C ' a b#62. dx dx 1 dx 1 dx' ' ' '" " 1 csc x sin x 1 sin x 1 (sin x 1) (sin x 1)sin x sin x 1 1 dx 1 sec x dx 1 sec x sec x tan x dx x tan x sec x C ' ' ' a b1 sin x sin x

cos x cos x # ## #

63. dx sin dx; sin dx 2 cos 2(cos cos 0)sin 0for 0 2' ' '0 0 02 2 21 1 1 1 cos x x x xx x# # # ##

#

#

!

11

1

( 2)( 2) 4

64. 1 cos 2x dx 2 sin x dx; 2 sin x dx 2 cos xsin x 0for 0 x' ' '

0 0 0

1 1 1 k k 1 1! 2 (cos cos 0) 2 2 165. 1 cos 2t dt 2 cos t dt; 2 cos t dt 2 sin tcos t 0for t

' ' '1 1 1

1 1 1

2 2 2 k k 1 11# #1

2 sin sin 2 1 1#66. 1 cos t dt 2 cos dt; 2 cos dt 2 2 sin cos 0for t 0

' ' ' 1 1 1

0 0 0

t t tt

# # ##

!

1 1

2 2 sin 0 sin 2 2 1#67. 1 cos d sin d ; sin d cos cos 0 cos ( )sin 0for 0' ' ' 1 1 1

0 0 0 k k c d # !) ) ) ) ) ) ) 1)1 ) 1 1 ( 1) 2

68. 1 sin d cos d ; cos d sin sin sin 1cos 0for ' ' '1 1 1

1 1 1

2 2 2 k k c d # # # #) ) ) ) ) ) ) 1

)

) 11

1

1

1

69. tan y 1 dy sec y dy; sec y dy ln sec y tan ysec y 0for y' ' '

1 1 1

1 1 1

4 4 4

4 4 4 k k c d k k# % % 1 1 1 14 4 ln 2 1 ln 2 1

70. sec y 1 dy tan y dy; tan y dy ln cos y lntan y 0for y 0' ' ' 1 1 14 4 4

0 0 0 k k c d k k # ! % " 1 14 2 ln 2 71. (csc x cot x) dx csc x 2 csc x cot x cot x dx 2 csc x 1 2 csc x cot x dx' ' '1 1 1

1 1 1

4 4 4

3 4 3 4 3 4 # # # #a b a b

2 cot x x 2 csc x 2 cot 2 csc 2 cot 2 csc c d $ %%11 1 1 1 1 1 13 3 34 4 4 4 4 4 2( 1) 2 2 2(1) 2 2 4 34 41 1 1#


72. (sec x 4 cos x) dx sec x 8 16 dx tan x 16x 4 sin 2x' '0 0

4 41 1 # # #

%!

c d1 cos 2x 1 tan 4 4 sin (tan 0 0 4 sin 0) 5 4 1 14 1 1#73. cos csc (sin ) d ; csc u du ln csc u cot u Cu sin du cos d' ') ) )

)

) ) k k

ln csc (sin ) cot (sin ) C k k) )74. 1 cot (x ln x) dx; cot u du ln sin u C ln sin (x ln x) Cu x ln xdu 1 dx' ' k k k k " "x

x

75. (csc x sec x)(sin x cos x) dx (1 cot x tan x 1) dx cot x dx tan x dx' ' ' ' ln sin x ln cos x C k k k k76. 3 sinh ln 5 dx sinh u du 6 cosh u C 6 cosh ln 5 Cu ln 52 du dx

' x x2 x2 ' ' #

77. ; 12 tan u C 12 tan y Cu y

du dy' '6 dy

y (1 y) 1 u2 y

12 du "" " #

78. ; sec u C sec 2x Cu 2xdu 2 dx' ' 'dx 2 dx du

x 4x 1 u u 12x (2x) 1 # ## " "

k k k k

79. ; 7 sec Cu x 1du dx' ' '7 dx 7 dx 7 du u

(x 1) x 2x 48 u u 49(x 1) (x 1) 49 7 7 " " # #

#

sec C " " x 780. ; sec u Cu 2x 1du 2 dx

' ' 'dx dx du(2x 1) 4x 4x (2x 1) (2x 1) 1 2u u 1

"#

" # ##

k k

sec 2x 1 C "#" k k

81. sec t tan (tan t) dt; tan u du ln cos u C ln sec u C ln sec (tan t) Cu tan tdu sec t dt' '#

# k k k k k k

82. csch C' dx xx x $

"$

"$#

83. (a) cos d (cos ) 1 sin d ; 1 u du u C sin sin Cu sin du cos d' ' '$ # # $") ) ) ) ) ) )

)

) )

a b a b u3 3$ (b) cos d (cos ) 1 sin d 1 u du 1 2u u du u u C' ' '& # # # % $# #) ) ) ) ) a b a b a b' 2 u3 5& sin sin sin C ) ) )23 5

$ &"

(c) cos d cos (cos ) d 1 sin (cos ) d' ' '* ) # %) ) ) ) ) ) ) ) a b a b84. (a) sin d 1 cos (sin ) d ; 1 u ( du) u Cu cos du sin d' ' '

$ # #) ) ) ) )

)

) )

a b a b u3$ cos cos C ) )" $3 (b) sin d 1 cos (sin ) d 1 u ( du) 1 2u u du' ' ' '& # # # %# #) ) ) ) ) a b a b a b cos cos cos C ) ) )23 5

$ &"

(c) sin d 1 u ( du) 1 3u 3u u du cos cos cos C' ' '( # # % ' $ &$) ) ) ) ) a b a b 3 cos5 7( )


(d) sin d sin (sin ) d 1 cos (sin ) d' ' '"$ "# # ') ) ) ) ) ) ) ) a b a b85. (a) tan d sec 1 (tan ) d sec tan d tan d tan tan d' ' ' '$ # # #"#) ) ) ) ) ) ) ) ) ) ) ) ) a b ' tan ln cos C "#

#) )k k

(b) tan d sec 1 tan d tan sec d tan d tan tan d' ' ' ' '& # $ $ # $ % $") ) ) ) ) ) ) ) ) ) ) ) ) a b a b 4 (c) tan d sec 1 tan d tan sec d tan d tan tan d' ' ' ' '( # & & # & ' &") ) ) ) ) ) ) ) ) ) ) ) ) a b a b 6 (d) tan d sec 1 tan d tan sec d tan d ;' ' ' '2k 1 2k 1 2k 1 2k 1 ) ) ) ) ) ) ) ) ) ) a b a b# # u du tan d u tan d tan tan du tan du sec d )) ) ) ) ) ) ) ) )# " "#' ' ' '2k 1 2k 1 2k 2k 1 2k 2k 1 2k k86. (a) cot d csc 1 (cot ) d cot csc d cot d cot cot d' ' ' ' '$ # # #"#) ) ) ) ) ) ) ) ) ) ) ) ) a b cot ln sin C "#

#) )k k

(b) cot d csc 1 cot d cot csc d cot d cot cot d' ' ' ' '& # $ $ # $ % $") ) ) ) ) ) ) ) ) ) ) ) ) a b a b 4 (c) cot d csc 1 cot d cot csc d cot d cot cot d' ' ' ' '( # & & # & ' &") ) ) ) ) ) ) ) ) ) ) ) ) a b a b 6 (d) cot d csc 1 cot d cot csc d cot d ;' ' ' '2k 1 2k 1 2k 1 2k 1 ) ) ) ) ) ) ) ) ) ) a b a b# # u du cot d u cot du cot du csc d )) ) ) ) ) )# "#' ' '2k 1 2k 1 2k 2k 1 k cot cot d "2k

2k 2k 1) ) )'

87. A (2 cos x sec x) dx 2 sin x ln sec x tan x '

1

1

4

4 c dk k 11

% %

2 ln 2 1 2 ln 2 1 2 2 ln 2 2 ln 22 1 2 12 1" #

2 2 ln 3 2 2

88. A (csc x sin x) dx ln csc x cot x cos x '1

1

6

2 c dk k 11

#'

ln 1 0 ln 2 3 ln 2 3 k k 3 3# #

89. V (2 cos x) dx sec x dx 4 cos x dx sec x dx ' ' ' '

1 1 1 1

1 1 1 1

4 4 4 4

4 4 4 41 1 1 1

# # # #

2 (1 cos 2x) dx tan x 2 x sin 2x [1 ( 1)] 1 1 1 1'

1

1

4

4 c d 11

1

1

% %

"#

%

%

2 2 2 1 2 1 1 1 1 1 1 1 14 4" "# # # #90. V csc x dx sin x dx csc x dx (1 cos 2x) dx ' ' ' '1 1 1 1

1 1 1 1

6 6 6 6

2 2 2 21 1 1

# # ##1

cot x x sin 2x 0 3 0 1 1c d 11

1 1 1 11

1

#' # # # # # #

" "#

' 63

3 1 1 1 1 1# 26 4 8 63 7 3


91. y ln (cos x) tan x sec x 1; L 1 dx dy dy dydx cos x dx dxsin x # ## # 'ab 1 sec x 1 dx sec x dx ln sec x tan x ln 2 3 ln 1 0 ln 2 3 ' '

0 0

3 31 1 a b c d k kk k # $!1

92. y ln (sec x) tan x sec x 1; L 1 dx dy dy dydx sec x dx dxsec x tan x # ## # 'ab sec x dx ln sec x tan x ln 2 1 ln 1 0 ln 2 1 '

0

41 c d k kk k 1%!93. M sec x (sec x) dx sec x dxx 4 4

4 4 ' '

1 1

1 1 " "# #

#

tan x 1 ( 1) 1; " "# #%

%c d c d1 1 M sec x dx ln sec x tan x '

1

1

4

4 c dk k 11

% %

ln 2 1 ln 2 1 ln 22 1" ln ln 3 2 2 ; x 0 by 2 1"##

symmetry of the region, and y MM ln 3 2 2x "

94. M csc x (csc x) dx csc x dxx 6 65 6 5 6

' '1 1

1 1

" "# #

#

cot x 3 3 3; " "# #& ''c d 11

M csc x dx ln csc x cot x '1

1

6

5 6 c dk k & ''11 ln 2 3 ln 2 3 ln 2 3

2 3

ln 2 ln 2 3 ; x by symmetry 2 34 3 ##

1

of the region, and y MM3

2 ln 2 3x

95. csc x dx (csc x)(1) dx (csc x) dx dx;' ' ' ' csc x cot x csc x csc x cot xcsc x cot x csc x cot x

#

ln u C ln csc x cot x Cu csc x cot xdu csc x cot x csc x dx a b k k k k # ' duu96. x 1 (x 1) (x 1)(x 1) (x 1) (x 1) (x 1) (x 1) (x 1)c d c da b # # #$ %$ # #$ #$#$ #$ (x 1) (x 1) 1 # #

#$ #$ x 1 2x 1 x 1

(a) x 1 (x 1) dx (x 1) 1 dx; udu dx' 'c da b # ##$ #$

""

2

x 1x 1

(x 1)#

(1 2u) du (1 2u) C 1 C C ' #$ "$# # # "$ "$3 3 2 3 x 1

x 1 x 1

(b) x 1 (x 1) dx (x 1) dx; u' 'c da b # ##$ #$ x 1 x 1x 1 x 1 k du k dx 2k dx; dx du x 1 x 1

x 1 (x 1) (x 1) k x 1(x 1) (x 1) (x 1) (x 1)

# k 1 k 1 c d#

#k 1

k 1

du; then, du du (x 1)2k x 1 x 1 k x 1 k x 1x 1 x 1 x 1 x 1 " " # #

#$# 1 k 1 k 1 3 k ' ' du u du (3k) u C u C C " " " # # # # #

"$

k x 1 k k x 1x 1 3 3 x 1' k 1 3k 1 1 3k 1 1 3k 1 3k '

Section 8.2 Integration by Parts 501

(c) x 1 (x 1) dx (x 1) dx;' c da b # ##$ #$ ' x 1x 1 du;

u tan xx tan udx

"

" "

#$ #$

ducos u

(tan u 1) tan u 1 cos u (sin u cos u) sin u cos utan u 1 du sin u cos u

#

# # #

' '

sin u cos u sin u sin u 2 sin cos usin u cos u sin u sin u 2 cos sin u

1 1 1

1 1 1

#

#

"

4 4

4 42 cos u cos

sin u'#

1

1

4

4u

#$

1

4 du

tan u sec u du tan u C C "# # # #$ # "$

"$' 1 1 14 4 4 1 tan u tan 3 3 tan u tan 1144

C 3 x 1x 1# "$

(d) u tan x tan u x tan u x dx 2 tan u du du ; " # " cos u cos u cos u

2 sin u 2d(cos u)# $ $

x 1 tan u 1 ; x 1 tan u 1 ; # # "sin u cos u 1 2 cos u cos u sin ucos u cos u cos u cos u

# # # # #

# # # #

(x 1) (x 1) dx' #$ %$ " ' a ba b a b1 2 cos u

cos u cos u

2d(cos u)cos u

#

#$

# #

#$ %$

$

1 2 cos u ( 2) cos u d(cos u) 1 2 cos u d 1 2 cos u ' 'a b a b a b# # ##$ #$"# 1 2 cos u C C C 3 3 3 x 1

x 1# # # # "$

"$

"$a b 1 2 cos u

cos u

cos u

#

#

"

#

(e) u tan tan u x 1 2(tan u 1) dx 2d(tan u); " # # x 1 x 1 2 ducos u# (x 1) (x 1) dx (tan u) (tan u 1) 2 2 d(tan u)' ' #$ %$ #$ %$ # 1 d 1 1 C 1 C " " " "# # #

#$ "$ "$' tan u 1 tan u 1 tan u 1 x 1

3 3 2

C 3 x 1x 1# "$

(f) u cos x

x cos u

dx sin u du

"

' 'sin u du sin u ducos u 1 (cos u 1) sin u 2 cos $ # ## %$ #$#

%$a b a b u

' ' 'du du du(sin u) 2 cos 2 sin cos cos

sin cos "$ #$# # #

%$ "$ &$

#

#

#

#

u u uu

u u"#

"$ tan d tan tan C tan C C ' "$ #$ ## # # # # # #

"$ "$ u u 3 u 3 u 3 cos u 1cos u 1

C 3 xx 1# " "$

(g) x 1 (x 1) dx; u cosh xx cosh udx sinh u

' c da b # #$"

' sinh u ducosh u 1 (cosh u 1)$ # ##a b

' ' 'sinh u du du dusinh u (cosh u 1) (sinh u) 4 cosh sinh cosh $ % # $ $% # #

a b "#u u u

tanh d tanh tanh C C C ' u u 3 u 3 cosh u 1 3 x 1cosh u 1 x 1# # # # # #

"$ #$ "$ "$

8.2 INTEGRATION BY PARTS

1. u x, du dx; dv sin dx, v 2 cos ; x x# # x sin dx 2x cos 2 cos dx 2x cos 4 sin C' 'x x x x x# # # # # 2. u , du d ; dv cos d , v sin ; ) ) 1) ) 1)"

1

cos d sin sin d sin cos C' ') 1) ) 1) 1) ) 1) 1) ) )1 1 1 1

" "#


3. cos t

t sin t#

2t cos t

2 sin t

0 t cos t dt t sin t 2t cos t 2 sin t C' # #

4. sin x

x cos x#

2x in xs

2 cos x

0 x sin x dx x cos x 2x sin x 2 cos x C' # #

5. u ln x, du ; dv x dx, v ; dx xx

#

#

x ln x dx ln x 2 ln 2 2 ln 2 ln 4' '1 1

2 2 x x dx x 3 3

x 4 4 4# # #

# #

# #

" "

6. u ln x, du ; dv x dx, v ; dx xx 4

$ %

x ln x dx ln x ' '1 1

e ee e

1 1

$ x x dx e x 3e 14 4 x 4 16 16% % % % %

7. u tan y, du ; dv dy, v y; " dy

1 y#

tan y dy y tan y y tan y ln 1 y C y tan y ln 1 y C' " " " # " #" # ' y dy1 ya b# a b

8. u sin y, du ; dv dy, v y; "

dy1 y #

sin y dy y sin y y sin y 1 y C' " " "

# ' y dy1 y # 9. u x, du dx; dv sec x dx, v tan x; #

x sec x dx x tan x tan x dx x tan x ln cos x C' '# k k

10. 4x sec 2x dx; [y 2x] y sec y dy y tan y tan y dy y tan y ln sec y C' # # ' ' k k 2x tan 2x ln sec 2x C k k11. ex

x e$

x

3x e#

x

6x e

x

6 e

x

0 x e dx x e 3x e 6xe 6e C x 3x 6x 6 e C' $ $ # $ #x x x x x x a b


12. ep

p e

% p

4p

e$ p

12p e# p

24p ep

24 e p

0 p e dp p e 4p e 12p e 24pe 24e C' % % $ # p p p p p p p 4p 12p 24p 24 e C a b% $ # p13. ex

x 5x e#

x

2x 5 e

x

2 e

x

0 x 5x e dx x 5x e (2x 5)e 2e C x e 7xe 7e C' a b a b# # # x x x x x x x x 7x 7 e C a b# x14. er

r r 1 e#

r

2r 1 e

r

2 e

r

0 r r 1 e dr r r 1 e (2r 1) e 2e C' a b a b# # r r r r r r 1 (2r 1) 2 e C r r 2 e C c d a ba b# #r r15. ex

x e&

x

5x e%

x

20x e$

x

60x e#

x

120x e

x

120 ee

x

0 x e dx x e 5x e 20x e 60x e 120xe 120e C' & & % $ #x x x x x x x x 5x 20x 60x 120x 120 e C a b& % $ # x


16. e4t

t e# "

44t

2t e

"16

4t

2 e

"64

4t

0 t e dt e e e C e e e C' # "#4t 4t 4t 4t 4t 4t 4t t 2t 2 t t4 16 64 4 8 3# #

e C t t4 8 3# "# 4t17. sin 2)

cos 2) )# "

2

2 sin 2) )

"4

2 cos 2

"8 )

0 sin 2 d cos 2 sin 2 cos 2'0

21) ) ) ) ) )#

# #"

#

! ) ) 1# 4

( 1) 0 ( 1) 0 0 1 1 1 1 1# # #8 4 4 4 8 8 4 " " " #18. cos 2x

x sin 2x$ "

2

3x cos 2x# "

4

6x sin 2x

"8

6 os 2x c

"16

0 x cos 2x dx sin 2x cos 2x sin 2x cos 2x'0

21$

#

#

! x 3x 3x 34 4 8$ # 1

0 ( 1) 0 ( 1) 0 0 0 1 1 1 1 1 1$ # # #16 16 8 8 8 16 4 163 3 3 3 3 3 3 4 a b

19. u sec t, du ; dv t dt, v ; " #

dt tt t 1 ##

t sec t dt sec t 2 ' ' '2 3 2 3 2 3

2 2 2

" "# #

#

# $ t t dt 2 t dt

t t 1 2 t 13 3 6# #

# # 1 1

t 1 3 1 3 5 5 4 5 59 9 3 9 3 9 3 93 3 5 3 31 1 1 1 1 " " "# # ## ## $

20. u sin x , du ; dv 2x dx, v x ; " # #

a b 2x dx1 x %

2x sin x dx x sin x x' ' '0 0 0

1 2 1 2 1 2 " # # " # #" #

!

"#

a b c da b 2x dx1 x 2 1 x6 d 1 x% %%1 1 x 1 1 1 11 1 4 1

3 6 3 12# # #

%" #

!

21. I e sin d ; u sin , du cos d ; dv e d , v e I e sin e cos d ; ' ') ) ) ) )) ) ) ) ) ) ) ) )c d u cos , du sin d ; dv e d , v e I e sin e cos e sin dc d ) ) ) ) ) ) ) )) ) ) ) )' e sin e cos I C 2I e sin e cos C I e sin e cos C, where C is ) ) ) ) ) )) ) ) ) ) )w w "# #a b a b Cw another arbitrary constant


22. I e cos y dy; u cos y, du sin y dy; dv e dy, v e ' y y yc d I e cos y e ( sin y) dy e cos y e sin y dy; u sin y, du cos y dy; y y y y' 'a b c dv e dy, v e I e cos y e sin y e cos y dy e cos y e sin y I Cd a b y y y y y y y' w 2I e (sin y cos y) C I e sin y e cos y C, where C is another arbitrary constant y y yw "# #a b Cw

23. I e cos 3x dx; u cos 3x; du 3 sin 3x dx, dv e dx; v e ' 2x 2x 2x "# I e cos 3x e sin 3x dx; u sin 3x, du 3 cos 3x, dv e dx; v e " "# # #

2x 2x 2x 2x3 ' I e cos 3x e sin 3x e cos 3x dx e cos 3x e sin 3x I C " " "# # # # #

w2x 2x 2x 2x 2x3 3 3 94 4 '

I e cos 3x e sin 3x C (3 sin 3x 2 cos 3x) C, where C C 13 3 e 44 4 13 13"# w w2x 2x2x

24. e sin 2x dx; [y 2x] e sin y dy I; u sin y, du cos y dy; dv e dy, v e' ' 2x y y y "# c d I e sin y e cos y dy u cos y, du sin y; dv e dy, v e "# c d y y y y' I e sin y e cos y e ( sin y) dy e (sin y cos y) I C " " "# # # w y y y y a b' 2I e (sin y cos y) C I e (sin y cos y) C (sin 2x cos 2x) C, where " "# w y y4 4e2x

C Cw#

25. e ds; e x dx xe dx; u x, du dx; dv e dx, v e ;3s 9 xds x dx' ' '3s 9 x x x x c d

#

23

2 23 3

xe dx xe e dx xe e C 3s 9 e e C2 2 2 23 3 3 3' a b x x x x x 3s 9 3s 9 ' 26. u x, du dx; dv 1 x dx, v (1 x) ; 23 $ x 1 x dx (1 x) x (1 x) dx (1 x)' '

0 0

1 1 2 2 2 2 43 3 3 5 15$ $"! ! "

27. u x, du dx; dv tan x dx, v tan x dx dx dx dx # # "' ' ' ' 'sin x cos x dxcos x cos x cos x

# #

# # #

tan x x; x tan x dx x(tan x x) (tan x x) dx 3 ln cos x ' '0 0

3 31 1 # $

! #

$

!c d k k 1 1 1 13 3 x#

3 ln ln 2 1 1 1 113 3 18 3 183 "# # #

28. u ln x x , du ; dv dx, v x; ln x x dx x ln x x x dx a b a b a b '# # # "(2x 1) dxx x x(x 1)2x# ' x ln x x x ln x x dx x ln x x 2x ln x 1 C a b a b a b k k# # # " ' '(2x 1) dx 2(x 1)x 1 x 1

29. sin (ln x) dx; du dx (sin u) e du. From Exercise 21, (sin u) e du e Cu ln x

dx e du'

" #x

sin u cos u

u

u u u' '

x cos (ln x) x sin (ln x) C "# c d


30. z(ln z) dz; du dz e u e du e u du;u ln z

dz e du' ' '# # #"

z

u u 2u

u

e2u

u e# "

22u

2u

e

"4

2u

2 e

"8

2u

0 u e du e e e C 2u 2u 1 C' # ## # "2u 2u 2u 2u u u e4 4# 2u c d

2(ln z) 2 ln z 1 C z4# c d#

31. (a) u x, du dx; dv sin x dx, v cos x; S x sin x dx [ x cos x] cos x dx [sin x]" ! ! ' '0 01 1

1 11 1

(b) S x sin x dx [ x cos x] cos x dx 3 [sin x] 3# # # ' '1 1

1 12 2 c d1 11 11 1 (c) S x sin x dx [ x cos x] cos x dx 5 [sin x] 5$ $ $# # ' '2 2

3 3

1 1

1 1

1 1

1 11 1

(d) S ( 1) x sin x dx ( 1) [ x cos x] [sin x]8" n 1 n 1 n 1 n 1n

n 1

n n

'1

1

1 1

1 1

c d ( 1) (n 1) ( 1) n ( 1) 0 (2n 1) n 1 n n 1 c d1 1 132. (a) u x, du dx; dv cos x dx, v sin x; S x cos x dx [x sin x] sin x dx [cos x] 2" # # ' '1 1

1 1

1 1

1 1

2 2

3 2 3 23 2 3 22 2 31 1 1

(b) S x cos x dx [x sin x] sin x dx [cos x] 4# # # ' '3 2 3 25 2 5 2

2 22 21 1

1 1

1 1

1 1

& &

$ $

5 31 1 1 (c) S x cos x dx [x sin x] sin x dx [cos x] 6$ # # ' '5 2 5 2

7 2 7 22 22 21 1

1 1

1 1

1 1

( (

& &

7 51 1 1 (d) S ( 1) x cos x dx ( 1) [x sin x] sin x dxn n n n2n 1 2 2n 1 2

2n 1 2 2n 1 2n 1 2

2n 1 2 ' '

#

1 1

1 1

1

1

( 1) ( 1) ( 1) [cos x] (2n 2n ) 2n n n n 1 n 1 22n 1 2 (2n 1) (2n 1) # # #"1 1 # 11 1 1 1 1 133. V 2 (ln 2 x) e dx 2 ln 2 e dx 2 xe dx ' ' '

0 0 0

ln 2 ln 2 ln 2x x x

1 1 1

(2 ln 2) e 2 xe e dx 1 1c d c d x x xln 2 ln 20 0 0ln 2' 2 ln 2 2 2 ln 2 e 2 ln 2 2 2 (1 ln 2) 1 1 1 1 1 c dx ln 20

34. (a) V 2 xe dx 2 xe e dx ' '0 0

1 1x x x

1 1 c d "!

2 e 2 1 1 1 c d " " ""!e e ex 2 1 4

e1


(b) V 2 (1 x)e dx; u 1 x, du dx; dv e dx, '0

1x x

1

v e ; V 2 (1 x) e e dx x x x0

11 c da b "! '

2 [0 1( 1)] e 2 1 1 1 1 c d x "! "e e21

35. (a) V 2 x cos x dx 2 [x sin x] sin x dx ' '0 0

2 21 1 1 1 1#!

2 [cos x] 2 0 1 ( 2) 1 1 1 1 1 11# ##!

(b) V 2 x cos x dx; u x, du dx; dv cos x dx, v sin x; '0

211 1 1# #

V 2 x sin x 2 sin x dx 0 2 [ cos x] 2 (0 1) 2 1 1 1 1 1 1 1 1# #! #!'0 21

36. (a) V 2 x(x sin x) dx; '0

1

1

sin x

x cos x#

2x in xs

2 cos x

0 V 2 x sin x dx 2 x cos x 2x sin x 2 cos x 2 4 1 1 1 1'0

1

# # #!c d a b1

(b) V 2 ( x)x sin x dx 2 x sin x dx 2 x sin x dx 2 [ x cos x sin x] 2 8 ' ' '0 0 0

1 1 1

1 1 1 1 1 1 1# # # $

!1 a b

8 1

37. (a) av(y) 2e cos t dt "#1 '02

t1

e " ##

!1

1 t sin t cos t (see Exercise 22) av(y) 1 e "#1 a b21

(b)

38. (a) av(y) 4e (sin t cos t) dt "#1'02

t1

e sin t dt e cos t dt 2 21 1' '

0 0

2 2t t1 1

e e 2 sin t cos t sin t cos t1

1 t t # #

#

!

e sin t 0 21

1c dt #!

(b)

39. I x cos x dx; u x , du nx dx; dv cos x dx, v sin x ' n n nc d" I x sin x nx sin x dx n n' "


40. I x sin x dx; u x , du nx dx; dv sin x dx, v cos x ' n n nc d" I x cos x nx cos x dx n n' "

41. I x e dx; u x , du nx dx; dv e dx, v e ' n ax n n ax axa

" " I e x e dx, a !x e n

a aax n axn ax ' "

42. I ln x dx; u ln x , du dx; dv dx, v x " ' a b a b n n n ln xx

a bn"

I x ln x n ln x dx a b a bn n' "

43. sin x dx x sin x sin y dy x sin x cos y C x sin x cos sin x C' '" " " " " a b44. tan x dx x tan x tan y dy x tan x ln cos y C x tan x ln cos tan x C' '" " " " " k k k ka b45. sec x dx x sec x sec y dy x sec x ln sec y tan y C' '" " " k k x sec x ln sec sec x tan sec x C x sec x ln x x 1 C " " " " #k ka b a b

46. log x dx x log x 2 dy x log x C x log x C' '2 2 2 2y 2 xln ln y

# #

47. Yes, cos x is the angle whose cosine is x which implies sin cos x 1 x ." " #a b 48. Yes, tan x is the angle whose tangent is x which implies sec tan x 1 x ." " #a b 49. (a) sinh x dx x sinh x sinh y dy x sinh x cosh y C x sinh x cosh sinh x C;' '" " " " " a b check: d x sinh x cosh sinh x C sinh x sinh sinh x dxc d a ba b " " " "

" x1 x 1 x # #

sinh x dx "

(b) sinh x dx x sinh x x dx x sinh x 1 x 2x dx' ' '" " " #" " #

"# a b1 x#

x sinh x 1 x C " # "#a b check: d x sinh x 1 x C sinh x dx sinh x dx a b" # " ""#

x x

1 x 1 x # #

50. (a) tanh x dx x tanh x tanh y dy x tanh x ln cosh y C' '" " " k k x tanh x ln cosh tanh x C; " "k ka b check: d x tanh x ln cosh tanh x C tanh x dxc dk ka b " " " " x1 x cosh tanh x 1 xsinh tanh x# " #"a ba b tanh x dx tanh x dx " " x x1 x 1 x# # (b) tanh x dx x tanh x dx x tanh x dx x tanh x ln 1 x C' ' '" " " " # # #" " x 2x1 x 1 x# # k k check: d x tanh x ln 1 x C tanh x dx tanh x dx k k" # " ""# x x1 x 1 x# #8.3 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

1. 5x 13 A(x 2) B(x 3) (A B)x (2A 3B)5x 13 A B(x 3)(x 2) x 3 x 2

B (10 13) B 3 A 2; thus, A B 52A 3B 13 5x 13 2 3(x 3)(x 2) x 3 x #

Section 8.3 Integration of Rational Functions by Partial Fractions 509

2. 5x 7 A(x 1) B(x 2) (A B)x (A 2B)5x 7 5x 7 A Bx 3x 2 (x 2)(x 1) x 2 x 1

#

B 2 A 3; thus, A B 5A 2B 7 5x 7 3 2x 3x 2 x 2 x 1 #

3. x 4 A(x 1) B Ax (A B) A 1 and B 3;A 1A B 4x 4 A B

(x 1) x 1 (x 1) # #

thus, x 4 1 3(x 1) x 1 (x 1) # #

4. 2x 2 A(x 1) B Ax ( A B) A 2A B 22x 2 2x 2 A B

x 2x 1 (x 1) x 1 (x 1)

# # #

A 2 and B 4; thus, 2x 2 2 4

x 2x 1 x 1 (x 1)

# #

5. z 1 Az(z 1) B(z 1) Cz z 1 (A C)z ( A B)z Bz 1 A B Cz (z 1) z z z 1

# ## #

B 1 A 2 C 2; thus, A C 0A B 1

B 1

z 2 2z (z 1) z z z 1

" " # #

6. 1 A(z 2) B(z 3) (A B)z (2A 3B)z A Bz z 6z z z 6 (z 3)(z 2) z 3 z $ # # #

" "

5B 1 B A ; thus, A B 02A 3B 1 " " 5 5 z z 6z z 3 z 2z$ # " "5 5

7. 1 (after long division); t 8 5t 2 5t 2 5t 2 A Bt 5t 6 t 5t 6 t 5t 6 (t 3)(t 2) t 3 t 2#

# # #

5t 2 A(t 2) B(t 3) (A B)t ( 2A 3B) B (10 2) 12A B 52A 3B 2

B 12 A 17; thus, 1 t 8 17 12t 5t 6 t 3 t 2#

#

8. 1 1 (after long division); t 9 9t 9 9t 9 9t 9 A B Ct Dt 9t t 9t t t 9 t t 9 t t t 9% # # #

% # % # # # # # # #

a b a b 9t 9 At t 9 B t 9 (Ct D)t (A C)t (B D)t 9At 9B # # # # $ #a b a b A 0 C 0; B 1 D 10; thus, 1

A C 0B D 9

9A 09B 9

t 9 10

t 9t t t 9%

% # # #

"

9. 1 A(1 x) B(1 x); x 1 A ; x 1 B ;" " " # #1 x 1 x 1 xA B# ln 1 x ln 1 x C' ' 'dx dx dx1 x 1 x 1 x # # #" " "# c dk k k k10. 1 A(x 2) Bx; x 0 A ; x 2 B ;" " " # #x 2x x x 2A B# ln x ln x 2 C' ' 'dx dx dx

x 2x x x 2# # # #" " " c dk k k k

11. x 4 A(x 1) B(x 6); x 1 B ; x 6 A ;x 4 A B 5 2 2x 5x 6 x 6 x 1 7 7 7

#

dx ln x 6 ln x 1 C ln (x 6) (x 1) C' ' 'x 4 2 dx 5 dx 2 5x 5x 6 7 x 6 7 x 1 7 7 7

"

#

k k k k k k12. 2x 1 A(x 3) B(x 4); x 3 B 7 ; x 4 A 9;2x 1 A B 7 9

x 7x 12 x 4 x 3 1 1

#

dx 9 7 9 ln x 4 7 ln x 3 C ln C' ' '2x 1 dx dxx 7x 12 x 4 x 3

(x 4)(x 3)

#*

(

k k k k


13. y A(y 1) B(y 3); y 1 B ; y 3 A ;yy 2y 3 y 3 y 1 4 4 4A B 1 3# " ln y 3 ln y 1 ln 5 ln 9 ln 1 ln 5' ' '

4 4 4

8 8 8y dy dy dyy 2y 3 4 y 3 4 y 1 4 4 4 4 4 4

3 3 3 3#

" " " ")

% k k k k

ln 5 ln 3 " "# # #ln 15

14. y 4 A(y 1) By; y 0 A 4; y 1 B 3 ;y 4y y y y 1 1A B 3 # dy 4 3 4 ln y 3 ln y 1 (4 ln 1 3 ln 2) 4 ln 3 ln ' ' '

1 2 1 2 1 2

1 1 1

y 4 dy dyy y y y 1

3 # #

""#

"#

c dk k k k ln ln ln ln 16 ln " " "8 16 8 8 8 4

27 27 27 15. 1 A(t 2)(t 1) Bt(t 1) Ct(t 2); t 0 A ; t 2" " #t t 2t t t 2 t 1A B C$ # B ; t 1 C ; 1 dt dt dt dt6 3 t t 2t t 6 t 2 3 t 1

" " " " #

' ' ' '$ #

ln t ln t 2 ln t 1 C " " "# k k k k k k6 316. (x 3) A(x 2)(x 2) Bx(x 2) Cx(x 2); x 0 A ; x 2x 3 A B C 32x 8x x x 2 x 8 " # # $ B ; x 2 C ; dx " " 16 16 2x 8x 8 x 16 x 2 16 x 2

5 x 3 3 dx dx 5 dx' ' ' '$

ln x ln x 2 ln x 2 C ln C 3 58 16 16 16 x(x 2) (x 2)k k k k k k " " &'

17. (x 2) (after long division); 3x 2 A(x 1) Bx 3x 2 3x 2 A Bx 2x 1 (x 1) (x 1) x 1 (x 1)$

# # # #

Ax (A B) A 3, A B 2 A 3, B 1; '0

1x dx

x 2x 1$

#

(x 2) dx 3 2x 3 ln x 1 ' ' '0 0 0

1 1 1dx dx xx 1 (x 1) x 1 #

""

!#

# k k 2 3 ln 2 (1) 3 ln 2 2 " "# #18. (x 2) (after long division); 3x 2 A(x 1) Bx 3x 2 3x 2 A B

x 2x 1 (x 1) (x 1) x 1 (x 1)$

# # # #

Ax ( A B) A 3, A B 2 A 3, B 1; '1

0x dx

x 2x 1$

#

(x 2) dx 3 2x 3 ln x 1 ' ' ' 1 1 1

0 0 0dx dx xx 1 (x 1) x 1 #

"!

"#

# k k 0 0 3 ln 1 2 3 ln 2 2 3 ln 2 " " " # #( 1) ( )19. 1 A(x 1)(x 1) B(x 1)(x 1) C(x 1) D(x 1) ;"

# # # #

a bx 1A B C D

x 1 x 1 (x 1) (x 1)# # # #

x 1 C ; x 1 D ; coefficient of x A B A B 0; constant A B C D " " $4 4 A B C D 1 A B ; thus, A B ; " " "# 4 4

dxx 1

' a b# # ln C " " " " " 4 x 1 4 x 1 4 (x 1) 4 (x 1) 4 x 1 2 x 1

dx dx dx dx x 1 x' ' ' '# # #

a b20. x A(x 1) B(x 1)(x 1) C(x 1); x 1x A B C(x 1) x 2x 1 x 1 x 1 (x 1)#

# # # #a b

C ; x 1 A ; coefficient of x A B A B 1 B ; " " # 2 4 4 (x 1) x 2x 13 x dx' ##a b

ln x 1 ln x 1 C " " " " # 4 x 1 4 x 1 2 (x 1) 4 4 (x 1)dx 3 dx dx 3' ' '#

k k k k C ln (x 1)(x 1)4 2(x 1)

k k "

$

21. 1 A x 1 (Bx C)(x 1); x 1 A ; coefficient of x1 A Bx C(x 1) x 1 x 1 x 1 # "# #a b# # a b A B A B 0 B ; constant A C A C 1 C ; 1 dx2 (x 1) x 1

"# '

0

1

a b#


dx ln x 1 ln x 1 tan x " " " " " # # # # " "

!2 x 1 x 1 4dx ( x 1)' '

0 0

1 1

#

k k a b ln 2 ln 2 tan 1 ln 1 ln 1 tan 0 ln 2 " " " " " " " "# # # # #" " 4 4 4 4 8( 2 ln 2)1 122. 3t t 4 A t 1 (Bt C)t; t 0 A 4; coefficient of t3t t 4 A Bt Ct t t t 1#

$ #

# # # a b A B A B 3 B 1; coefficient of t C C 1; dt '

1

3 3t t 4t 1#

$

4 dt 4 ln t ln t 1 tan t ' '1 1

3 3 dtt t 1 2

( t 1)

" # " $

"# k k a b

4 ln 3 ln 4 tan 3 4 ln 1 ln 2 tan 1 2 ln 3 ln 2 ln 2 " " "# # #" " 1 13 4 2 ln 3 ln 2 ln "# # #

1 1

1 19

2

23. y 2y 1 (Ay B) y 1 Cy Dy 2y 1 Ay B Cy Dy 1 y 1y 1#

# #

# ##

# #

a b a b a b Ay By (A C)y (B D) A 0, B 1; A C 2 C 2; B D 1 D 0; $ # dy dy 2 dy tan y C' ' 'y 2y 1 y

y 1 y 1y 1 y 1#

# #

# ## #

" "

"a b a b

24. 8x 8x 2 (Ax B) 4x 1 Cx D8x 8x 2 Ax B Cx D4x 1 4x 14x 1#

# #

# ##

# #a b a b a b

4Ax 4Bx (A C)x (B D); A 0, B 2; A C 8 C 8; B D 2 D 0; $ # dx 2 8 tan 2x C' ' '8x 8x 2 dx x dx

4x 1 4x 14x 1 4x 1#

# #

# ## #

"

"a b a b

25. 2s 22s 2 As B C D Es 1 (s 1) s 1 s 1 (s 1) (s 1)

a b# $ # # $

(As B)(s 1) C s 1 (s 1) D s 1 (s 1) E s 1 $ # # # #a b a b a b As ( 3A B)s (3A 3B)s ( A 3B)s B C s 2s 2s 2s 1 D s s s 1 c d a b a b% $ # % $ # $ # E s 1 a b# (A C)s ( 3A B 2C D)s (3A 3B 2C D E)s ( A 3B 2C D)s ( B C D E) % $ #

summing all eq

A C 03A B 2C D 0

3A 3B 2C D E 0 A 3B 2C D 2 B C D E 2

uations 2E 4 E 2;

summing eqs (2) and (3) 2B 2 0 B 1; summing eqs (3) and (4) 2A 2 2 A 0; C 0 from eq (1); then 1 0 D 2 2 from eq (5) D 1; ds 2 (s 1) (s 1) tan s C' ' ' '2s 2 ds ds ds

s 1 (s 1) s 1 (s 1) (s 1)

# " "a b# $ # # $

26. s 81 A s 9 (Bs C)s s 9 (Ds E)ss 81 A Bs C Ds Es s 9 s 9s s 9%

# #

# ##

% # ##

a b a b a b a b A s 18s 81 Bs Cs 9Bs 9Cs Ds Es a b a b% # % $ # # (A B)s Cs (18A 9B D)s (9C E)s 81A 81A 81 or A 1; A B 1 B 0; % $ # C 0; 9C E 0 E 0; 18A 9B D 0 D 18; ds 18 ' ' 's 81 ds s ds

s s 9 s 9s%

# #

# #

a b a b

ln s C k k 9s 9a b#

27. 2 5 8 4 (A B) 2 2 C D2 5 8 4 A B C D 2 2 2 2 2

) ) ) ) )

) ) ) )) )

$ #

# #

# ##

#

$ # #a b a b ) ) ) ) ) ) )a b

A (2A B) (2A 2B C) (2B D) A 2; 2A B 5 B 1; 2A 2B C 8 C 2; ) ) )$ # 2B D 4 D 2; d d d ' ' '2 5 8 4 2 1 2 2

2 2 2 2 2 2) ) ) ) )

) ) ) )) )

$ #

# #

# ##

a b a ba b) ) )

d ' ' ' ' '2 2 d d 2 2 2 2 ( 1) 1 2

d 2 2 d 2 2 2 2

) ) )

) ) ) ) ) ) ) ) )

) ) ) )

) )

" # # #

# # # # #

# #

#

#

)a b a b

a b


ln 2 2 tan ( 1) C " # # ") )# 2 a b) ) )28. 4 2 3 1) ) ) ) ) ) )

) ) ))

% $ #

# # #

$ # $#

4 2 3 1 A B C D E F 1 1 1 1

% $ #a b a b a b ) ) ) )

(A B) 1 (C D) 1 E F (A B) 2 1 C D C D E F ) ) ) ) ) ) ) ) ) ) ) )a b a b a b a b# # % # $ ## A B 2A 2B A B C D C D E F a b a b) ) ) ) ) ) ) ) )& % $ # $ # A B (2A C) (2B D) (A C E) (B D F) A 0; B 1; 2A C 4 ) ) ) ) )& % $ # C 4; 2B D 2 D 0; A C E 3 E 1; B D F 1 F 0;

d 4 tan 2 1 1 C' ' ' ') ) ) ) ) ) ) ) )) ) ))

% $ #

# # #

$ # $#

4 2 3 1 d d d 1 1 1 1 4

"

" # #" #

a b a b a b) ) ) ) a b a b29. 2x 2x ; 1 A(x 1) Bx; x 0 A 1;2x 2x 1 A B

x x x x x(x 1) x(x 1) x x 1$ #

# #

" " "

x 1 B 1; 2x dx x ln x ln x 1 C x ln C ' ' ' '2x 2x 1 dx dx x 1x x x x 1 x$ #

#

# #k k k k 30. x 1 x 1 ; 1 A(x 1) B(x 1);x A B

x 1 x 1 (x 1)(x 1) (x 1)(x 1) x 1 x 1%

# # # #" " " a b a b

x 1 A ; x 1 B ; dx x 1 dx " " " "# # # # #' ' ' 'x dx dx

x 1 x 1 x 1%

#

a b x x ln x 1 ln x 1 C x ln C " " " " $ # # # 3 3 x 1

x x 1k k k k $

31. 9 (after long division); 9x 3x 1 9x 3x 9x 3x 1 A B Cx x x (x 1) x (x 1) x x x 1$ # #

$ # # # #

"

9x 3x 1 Ax(x 1) B(x 1) Cx ; x 1 C 7; x 0 B 1; A C 9 A 2; # # dx 9 dx 2 7 9x 2 ln x 7 ln x 1 C' ' ' ' '9x 3x 1 dx dx dx

x x x x x 1 x$

$ # #

" k k k k

32. (4x 4) ; 12x 4 A(2x 1) B16x 12x 4 12x 4 A B4x 4x 1 4x 4x 1 (2x 1) 2x 1 (2x 1)$

# # # #

A 6; A B 4 B 2; dx 4 (x 1) dx 6 2 ' ' ' '16x dx dx4x 4x 1 2x 1 (2x 1)$

# #

2(x 1) 3 ln 2x 1 C 2x 4x 3 ln 2x 1 (2x 1) C, where C 2 C # # "" " "k k k k2x 133. y ; 1 A y 1 (By C)y (A B)y Cy Ay y 1 By Cy y y y 1 y y 1 y y 1A% #

$ # # #

" " # # a b a b a b A 1; A B 0 B 1; C 0; dy y dy ' ' ' 'y y 1 dy y dyy y y y 1% #

$ #

ln y ln 1 y C y#

# #" #k k a b

34. 2y 2 ; 2y By Cy y y 1 y y y 1 y y y 1 y 1 (y 1) y 1 y 12 2 2 A%$ # $ # $ # # #

a b 2 A y 1 (By C)(y 1) Ay A By Cy By C (A B)y ( B C)y (A C) a b a b a b# # # # A B 0, B C 0 or C B, A C A B 2 A 1, B 1, C 1;

dy 2 (y 1) dy dy' ' ' ' '2y dy y dyy y y 1 y 1 y 1 y 1%

$ # # #

(y 1) ln y 1 ln y 1 tan y C y 2y ln y 1 ln y 1 tan y C, # # " # # "" "# #"k k a b k k a b where C C 1 "

35. e y ln C ln C' ' ' 'e dt e 1e 3e 2 y 3y 2 y 1 y 2 y 2 e

dy dy dy y 1t t2t t t #

c d t#

36. dt e dt; dy y dyy edy e dt' ' ' ' 'e 2e e e 2e 1

e 1 e 1 y 1 y 1 2 y 1y 2y 1 y 1 y y4t 2t t 3t t

2t 2t

3 2

tt

t # # # dy ' dyy 1# ln y 1 tan y C e ln e 1 tan e C y2 2

2t 2t t2 " " "# #

# " "a b a b a b


37. ; [sin y t, cos y dy dt] dt ln C' ' 'cos y dy dysin y sin y 6 t t 6 5 t 2 t 3 5 t 3

t 2# #

" " " " ln C " 5 sin y 3

sin y 2 38. ; cos y ln C ln ' ' ' 'sin d cos 2

cos cos 2 y y 2 3 y 2 3 y 1 3 y 1 3 cos 1dy dy dy y 2) ) )

) ) )# # " " " " c d ) C

ln C ln C " " 3 1 cos 3 cos 22 cos cos 1 ) )

) )

39. dx dx 3 dx' ' '(x 2) tan (2x) 12x 3x tan (2x)4x 1 (x 2) 4x 1 (x 2)x

# " $ "

# # # #a b tan (2x) d tan (2x) 3 6 3 ln x 2 C "# " "' ' 'a b k kdx dx 6x 2 (x 2) 4 x 2tan 2x# " #a b

40. dx dx dx' ' '(x 1) tan (3x) 9x x tan (3x)9x 1 (x 1) 9x 1 (x 1)x

# " $ "

# # # #a b tan (3x) d tan (3x) ln x 1 C " " " 3 x 1 (x 1) 6 x 1dx dx 1tan 3x' ' 'a b k k# " #a b

41. t 3t 2 1; x ln C; Ce ; t 3 and x 0a b # dx dt dt dt t 2 t 2dt t 3t 2 t 2 t 1 t 1 t 1 x' ' '# C e x ln 2 ln t 2 ln t 1 ln 2 " " # #

t 2 t 2t 1 t 1

x k k k k42. 3t 4t 1 2 3; x 2 3 3 3a b % #

dx dt dt dtdt t 13t 4t 1 t

' ' '% #

#

" #

3

3 tan 3t 3 tan t C; t 1 and x C C " " 1 1 3 3 34 4 41 1 1 x 3 tan 3t 3 tan t " " 143. t 2t 2x 2; ln x 1 ln x 1 ln C;a b k k k k' ' # " " " "# # # # # dx dx dt dt dt tdt x 1 t 2t t t 2 t ' ' # t 1 and x 1 ln 2 ln C C ln 2 ln 3 ln 6 ln x 1 ln 6 x 1 " #3 t 2 t

t 6tk k x 1, t 0 6tt #

44. (t 1) x 1 tan x ln t 1 C; t 0 and x tan ln 1 C dx dx dtdt x 1 t 1 4 4# " " ' '# k k k k1 1 C tan 1 tan x ln t 1 1 x tan (ln (t 1) 1), t 1 " "14 k k45. V y dx dx 3 dx 3 ln 3 ln 25 1 1 1 1 1' ' '

0 5 0 5 0 5

2 5 2 5 2 5

#

" " #&

!&9 x

3x x x 3 x x 3#

46. V 2 xy dx 2 dx 4 dx 1 1 1' ' '0 0 0

1 1 12x 2

(x 1)(2 x) 3 x 1 3 2 x " " "

ln x 1 2 ln 2 x (ln 2) a bk k k k4 43 31 1"!47. A tan x dx x tan x dx ' '

0 0

3 330

" "c d x1 x#

ln x 1 ln 2; 1 1 3 33 3 a b"# # 30 x x tan x dx " "A '0

3

x tan x dx " " "# # # "

A 1 xx 30 0 3' # #

x tan x " "# #"

A a b1 30 1.10 " "# # #A 6 A 3

3 32 1 1 1


48. A dx 3 2 3 ln x ln x 3 2 ln x 1 ln ; ' ' ' '3 3 3 3

5 5 5 54x 13x 9 dx dx dx 125x 2x 3x x x 3 x 1 9#

$ #

&$c dk k k k k k

x dx 4x 3 2 (8 11 ln 2 3 ln 6) 3.90 " " " & $A x 2x 3x A x 3 x 1 Ax 4x 13x 9 dx dx' ' '3 3 35 5 5a b#$ #

c d49. (a) kx(N x) k dt k dt ln kt C;dx dx dx dx xdt x(N x) N x N N x N N x ' ' ' ' ' " " " k , N 1000, t 0 and x 2 ln C ln ln " " " " "250 1000 998 1000 1000 x 250 1000 499

2 x t ln 4t e 499x e (1000 x) 499 e x 1000e x a b499x 499x 1000e1000 x 1000 x 499 e4t 4t 4t 4t 4t4t (b) x N 500 500 500 499 500e 1000e e 499 t ln 499 1.55 days " "# 1000e499 e 44t 4t 4t

4t4t

50. k(a x)(b x) k dtdx dxdt (a x)(b x) (a) a b: k dt kt C; t 0 and x 0 C kt ' 'dx(a x) a x a a x a " " " "# a x x a " " a x a akt 1 akt 1 akt 1

akt a a a kt#

(b) a b: k dt k dt ln kt C; ' ' ' ' 'dx dx dx b x(a x)(b x) b a a x b a b x b a a x " " " t 0 and x 0 ln C ln (b a)kt ln e " b a a a x a a x ab b x b b x b b a kt x

ab 1 ea be

b a kt

b a kt

51. (a) dx x 4x 5x 4x 4 dx' '0 0

1 1x (x 1)

x 1 x 1 74 22% %# #

' & % # 1 (b) 100% 0.04%227 1

1

(c) The area is less than 0.003

52. P(x) ax bx c, P(0) c 1 and P (0) 0 b 0 P(x) ax 1. Next, # w # ; for the integral to be a function, we must have A 0 andax 1 A B C D E

x (x 1) x x x x 1 (x 1)#

$ # # $ #

rational

D 0. Thus, ax 1 Bx(x 1) C(x 1) Ex (B E)x (C 2B)x (B 2C)x C # # # $ $ #

E B; x 1 a 1 E; therefore, 1 2B a 1 2E a 1 2(a 1) aB E 0

C 2B aC

"

a 3

Section 8.4 Trigonometric Integrals 515

8.4 TRIGONOMETRIC INTEGRALS

1. sin x dx sin x sin x dx cos x sin x dx 2cos x cos x sin x dx' ' ' '0 0 0 0

1 1 1 1/2 /2 /2 /25 4 " " a b a b a b# # ## #

sin x dx 2cos x sin x dx cos x sin x dx cos x # ' ' '0 0 0 0

1 1 1 1/2 /2 /24 cos x cos x

3 5

/2# 3 5

! " a b # " )$ & "& 2. sin dx (using Exercise 1) sin dx 2cos sin dx cos sin dx' ' ' '

0 0 0 0

1 1 1 15 4x x x x x x

2 2 2 2 2 2 #

cos cos cos # ! # a bx x x2 3 2 5 23 5% # % # "'$ & "&10 3. cos x dx cos x cos x dx sin x cos x dx cos x dx sin x cos x dx' ' ' ' '

# # #

1 1 1 1 1

1 1 1 1 1

/2 /2 /2 /2 /2

/2 /2 /2 /2 /23 " a b a b

sin x " " sin x33 11

#

#

" " %$ $ $

4. 3cos 3x dx cos 3x cos 3x 3dx sin 3x cos 3x 3dx sin 3x sin 3x cos 3x 3dx' ' ' '0 0 0 0/6 /6 /6 /6

51 1 1 1

" " # a b a b a b# # # %# # cos 3x 3dx sin 3x cos 3x 3dx sin 3x cos 3x 3dx sin 3x # # ' ' '0 0 0

/6 /6 /6sin 3x sin 3x

3 5

61 1 1 1# %

3 50

" ! a b2$ & "&" )

5. sin y dy sin y sin y dy cos y sin y dy sin y dy cos y sin y dy' ' ' ' '0 0 0 0 0/2 /2 /2 /2 /2

7 6 2 21 1 1 1 1

" $a b$ cos y sin y dy cos y sin y dy cos y $ $ $ ! " " ' '0 0

/2 /24 6 cos y cos y cos y

3 51 1 1 a b 3 & (( & ( $ $ " "'0

6. 7cos t dt (using Exercise 5) 7 cos t dt sin t cos t dt sin t cos t dt sin t cos t dt' ' ' ' '0 0 0 0 0/2 /2 /2 /2 /2

7 2 4 61 1 1 1 1

$ $ 7 sin t 7 7 $ $ " " ! a bsin t sin t sin t3 53 & (( & ( $ " "'10 7. sin x dx dx cos x cos x dx dx cos x dx dx' ' ' ' ' '

0 0 0 0 0 0

1 1 1 1 1 1

) ) # " # # # # # # # #4 cos x cos 4x a b" # "# ## # x sin x dx cos 4x dx x sin 4x # # # # # $c d 1 1 1 10 0 0 0' ' 1 1 1 1"# 8. cos 2 x dx dx cos 4 x cos 4 x dx dx cos 4 x dx dx' ' ' ' ' '

0 0 0 0 0 0

1 1 1 1 1 1) ) # " # # % #4 cos 4 x cos x1 1 1 1 a b" " )# ## #1 1

x sin 4 x dx cos x dx x sin x # ) # ) # " $ " "" ")1 11 1 10 0 01 1 0' '

9. 16 sin x cos x dx 16 dx cos x dx dx dx' ' ' ' '

# # #" # " # "# # #1 1 1 1 1

1 1 1 1 1

/4 /4 /4 /4 /4

/4 /4 /4 /4 /4cos x cos x cos 4x % " # % % a b

4x dx cos 4x dx 2x =2 # # c d 11 1 1

1 11

1

1 1/4/4 /4 /4

/4 /4sin 4x

2/4

/4 # #' ' 1 1 1 1

10. 8 sin y cos y dy 8 dy dy cos y dy cos y dy cos y dy' ' ' ' ' '0 0 0 0 0 0

1 1 1 1 1 14 cos y cos y# # $" # " #

# #

# # # #

y sin 2y dy sin y cos y dy dy cos 4y dy cos y dy " # # # a b" " "" # # ##2 cos 4y1 1 1 1 1 10 0 0 0 0 0' ' ' ' '1 sin y cos y dy y sin 4y sin 2y # # '

0 0

1 11 1# " " " "

# ) # # # #1 1 sin 2y33


11. 35 sin x cos x dx 35 sin x sin x cos x dx 35 sin x cos x dx 35 sin x cos x dx' ' ' '0 0 0 0

1 1 1 1/2 /2 /2 /24 3 4 4 6 " a b#

35 35 7 5 ! # a b a bsin x sin x5 7 /25 7 1012. cos 2x sin 2x dx'

0 0

1 12 cos 2x

3 ! " " "# ' '3

13. 8cos 2 sin 2 d 8 cos 2'0 0

0

1)

11

/43 4cos 2

4

/4 /4) ) ) ) ! " " c d a b a b"# 4

14. sin 2 cos 2 d sin 2 sin 2 cos 2 d sin 2 cos 2 d sin 2 cos 2 d' ' ' '0 0 0 0

1 1 1 1/2 /2 /2 /22 3 2 2 2 4) ) ) ) ) ) ) ) ) ) ) ) ) " a b

! " "# #sin 2 sin 23 5 /23 5) ) 1015. dx sin dx sin dx cos ' ' '

0 0 0 0

2 2 2cos x x x x 2

1 1 11 "# # # # # # # %

16. cos 2x dx sin 2x dx sin 2x dx cos 2x' ' '0 0 0 0

1 1 1 1 " # l l # # # # # #

17. sin t dt cos t dt cos t dt cos t dt sin t sin t' ' ' '0 0 0 0

1 1 1 1

1

1 1

1 c d c d" l l " ! ! " ## /2 /2 /2 /2

18. cos d sin d sin d cos ' ' '0 0 0 0

1 1 11 c d" l l " " ##) ) ) ) ) ) )

19. tan x dx sec x dx sec x dx ln sec x tan x ln ln' ' '

#1 1 1

1 1 11

1/4 /4 /4

/4 /4 /4 /4/4

c d " l l l l # " # " ln ln # " # #"#"20. sec x dx tan x dx tan x dx tan x dx ln sec x ln sec x ' ' ' '

!#

!!

!1 1 1

1 1 11

/4 /4 /4

/4 /4 /4 /4 c d c d " l l l l l l1/4

ln ln ln ln ln ln " # # " # # #a b a b 21. cos 2 d sin d sin d cos sin ' ' '

0 0 0 0

1 1 11

/2 /2 /2 /2) ) ) ) ) ) ) ) ) ) ) ) c d a b" # l l # # # " #

22. cos t dt sin t dt sin t dt sin t dt sin t dt cos t sin t dt' ' ' ' ' ' !

# # $ $ $ #$# $#! !

1 1 1 1 1

1 1 1 1a b a b a b " " cos t sin t dt sin t dt cos t sin t dt sin t dt cos t sin t dt cos t " ' ' ' ' '

! ! !# # #

! ! !

1 1 1

1 1 1

a b cos t33 cos t " " " " cos t33 1! " " " " )$ $ $ $ $23. 2 sec x dx; u sec x, du sec x tan x dx, dv sec x dx, v tan x;'

!$ #

1/3

2 sec x dx 2 sec x tan x sec x tan x dx sec x sec x dx' ' '

! ! !$ !

$1 1 11/3 /3 /32 2 # # " ! # # $ # "c d a b

sec x dx sec x dx; 2 2 sec x dx ln sec x + tan x % $ # # % $ # l l c d' ' '

! ! !$ $ !

$1 1 1 1/3 /3 /3

2 2 sec x dx ln + ln ln '

!$

1/3 % $ # l " !l # l # $ l % $ # # $

2 sec x dx ln '

!$

1/3 # $ # $

Section 8.4 Trigonometric Integrals 517

24. e sec e dx; u sec e , du sec e tan e e dx, dv sec e e dx, v tan e .' x x x x x x x x x$ #a b a b a b a b a b a b e sec e dx sec e tan e sec e tan e e dx' 'x x x x x x x$ #a b a b a b a b a b sec e tan e sec e sec e e dx "a b a b a ba ba bx x x x x' # sec e tan e sec e e dx sec e e dx a b a b a b a bx x x x x x' '$ e sec e dx sec e tan e ln sec e tan e C# ' x x x x x x$a b a b a b a b a b e sec e dx sec e tan e ln sec e tan e C' x x x x x x$ "#a b a b a b a b a b 25. sec d tan sec d sec d tan sec d tan ' ' ' '

! ! ! !

!

1 1 1 1)

1/4 /4 /4 /44 2 2 2 2 2 tan

3

4) ) ) ) ) ) ) ) ) ) ) " a b 3

" ! a b" %$ $26. 3sec 3x dx tan 3x sec 3x 3dx sec 3x 3dx tan 3x sec 3x 3dx' ' ' '

! ! ! !

1 1 1 1/12 /12 / /124 2 2 2 2 2a b a b a b a b a b a ba b "

tan 3x " ! a b a b tan 3x3 123a b 1! " %$ $27. csc d cot csc d csc d cot csc d cot' ' ' '

1 1 1 1

1 1 1 1)

1

1/4 /4 /4 /4

/2 /2 /2 /24 cot

3

2

/4) ) ) ) ) ) ) ) ) ) ) " a b # # # # # $

! " a b " %$ $28. csc d cot csc d csc d cot csc d cot' ' ' '

1 1 1 1

1 1 1 1) ) ) ) ) ) )

1

1/2 /2 /2 /24 cot

3 2$ $ " $ $ ' '# # # # # # #

# # # # #

) ) ) ) $ #)

' ! # ! ' " # " )a b a b29. 4 tan x dx 4 sec x tan x dx 4 sec x tan x dx 4 tan x dx ln sec x' ' ' '0 0 0 0

/4 /4 /4 /43 tan x

41 1 1 1 1 " % % l la b # # # !#

2 ln ln ln " % # # ! % " # # #a b 30. 6 tan x dx 6 sec x tan x dx 6 sec x tan x dx 6 tan x dx' ' ' '

# #

1 1 1 1

1 1 1 1

/4 /4 /4 /4

/4 /4 /4 /44 2 2 2 " a b

6 sec x tan x dx 6 sec x 1 dx 6 sec x dx 6 dx ' ' ' ' ' $

#1 1 1 1

1 1 1 11

1/4 /4 /4 /4

/4 /4 /4 /42 2 2 tan x

4

4a b $

tan x x # " " ' ' % ' " " $ )a b c d c d a ba b a b1 11 1

1 1

$ $# #

4 44 4 1

31. cot x dx csc x cot x dx csc x cot x dx cot x dx ln csc x' ' ' '1 1 1 1

1 1 1 1 1

1/6 /6 /6 /6

/3 /3 /3 /33 2 2 cot x

3

6 " l la b ##

ln ln ln $ # $" " # %# $ $$ 32. 8 cot t dt 8 csc t cot t dt 8 csc t cot t dt 8 cot t dt' ' ' '

1 1 1 1

1 1 1 1

/4 /4 /4 /4

/2 /2 /2 /24 2 2 2 2 2 " a b

8 8 csc t dt cot t t " ! " ) ) ) ! " % # # a b a b c d c d a bcot t3 324 /4/2

2 2 24 4

3 1

1 1

11 1

1 1

) ) "' $ $

' 1 1 1

33. sin 3x cos 2x dx sin x sin 5x dx cos x cos 5x' '

! !" " " " " " '# # & # & & &

!

1 1 1 " " a b

34. sin 2x cos 3x dx sin x sin 5x dx cos x cos 5x' '! !

" " " " " " ## # & # # & &

!

1 11

2 2 2 ! " a b a b a ba b


35. sin 3x sin 3x dx cos cos 6x dx dx cos 6x dx x sin 6x' ' ' '

" " " " "# # # # "# # #1 1 1 1

1 1 1 11

1

1 1 ! ! a b 136. sin x cos x dx sin sin 2x dx sin 2x dx cos 2x' ' '

! ! !

" " " " "# # % % #

#!

1 1 11

2 2 2 ! " " a b c d a b

37. cos 3x cos 4x dx cos x cos 7x dx sin x sin 7x 0 0' '! !

" " " "# ( !

1 11

a b a b a ba b 2 238. cos 7x cos x dx cos 6x cos 8x dx sin 6x sin 8x 0' '

" " " "#

1 1

1 11

12 2

2 2

2 6 82

2 a b 39. x t t x ; y y ; t x 2 ; ! # ! #$ # $ #$# #

t x# $

A x dx; u du uu xdu 9x dx

# " " "

' '! !# % $

* #% %%

$

* $#

*

!

2 2x

9

9 9

2#$ #$$

#$

1 a b1 1 2 " * "##(

#$ $#1

40. y ln cos x ; y tan x; y tan x; tan x dx sec x dx ln sec x tan x " l l l la b a b c dw w # #! !

#

!sin x

cos x

3 3 /3' '1 1 1

ln 2 ln ln 2 $ " ! $ a b

41. y ln sec x ; y tan x; y tan x; tan x dx sec x dx ln sec x tan x " l l l la b a b c dw w ##! !

#

!sec x tan x

sec x

4 4 /4' '1 1 1

ln ln ln # " ! " # " a b

42. M sec x dx ln sec x tan x ln ln ln l l # " l # "l '

#"

#"1

11

14

4 /44c d

y dx tan x " " " " " "

#

ln ln ln ln

#" #" #" #"

#" #" #" #"

'1

11

14

4sec x 4

4#

# #

c d a ba b x, y ln !a b #"#" "

43. V sin x dx dx dx cos 2x dx x sin 2x ! ! ! 1 1 1' ' ' '! ! ! !

# "# # # # #! !

1 1 1 11 1 1 1 1 1 11 1cos 2x

2 4 4c d c d a b a b #

44. A cos 4x dx cos 2x dx cos 2x dx cos 2x dx cos 2x dx " # l l # # #' ' ' ' '! ! ! % $ %

% $ %1 1 1 1 1

1 1

sin 2x sin 2x sin 2x " ! " " ! " # # # # # # # # # ## # # # # #

% $ %! % $ %

c d c d c d a b a b a b 1 11

1

1

45. (a) m n m n and m n sin mx sin nx dx cos m n x cos m n x dx# ## #

"# ! !

' 'k k

k k1 1 c da b a b sin m n x sin m n x " " "#

# a b a bm n m n

k 1k

sin m n k sin m n k sin m n k sin m n k # # " " " " " "# # a b a b a b a ba ba b a ba b a b a bm n m n m n m n1 1 sin m n k sin m n k sin m n k sin m n k !" " " "# # # # a b a b a b a bm n m n m n m na b a b a b a ba b a b a b a b sin mx and sin nx are orthogonal.

(b) Same as part since cos dx . m n m n and m n cos mx cos nx dx"## #

# #' 'k k

k k1 1! ! ! 1

cos m n x cos m n x dx sin m n x sin m n x " " " "# # # #'

k

k

m n m n

k1 1c d a b a ba b a b k

sin m n k sin m n k sin m n k sin m n k # # " " " "# # # # a b a b a b a bm n m n m n m na b a b a b a ba ba b a ba b a b a b1 1 sin m n k sin m n k sin m n k sin m n k !" " " "# # # # a b a b a b a bm n m n m n m na b a b a b a ba b a b a b a b

Section 8.5 Trigonometric Substitutions 519

cos mx and cos nx are orthogonal.

(c) Let m n sin mx cos nx sin sin m n x and sin dx and sin m n x dx 0 ! ! ! " " "# # ## #a b a ba b a ba b ' 'k kk k1 1

sin mx and cos nx are orthogonal if m n. Let m n.

sin mx cos nx dx sin m n x sin m n x dx cos m n x cos m n x' 'k kk k

m n m n

k# #" " " "# #

#1 1 1 c d a b a ba b a b

k

cos m n k cos m n k cos m n k cos m n k # # " " " "# # # # a b a b a b a bm n m n m n m na b a b a b a ba ba b a ba b a b a b1 1 cos m n k cos m n k cos m n k cos m n k !" " " "# # # # a b a b a b a bm n m n m n m na b a b a b a ba b a b a b a b sin mx and cos nx are orthogonal.

46. f x sin mx dx sin nx sin mx dx. Since sin nx sin mx dx ,for m n1 for m n" "

"

1 1 11 1 1

1 1 1' ' 'a b ! ! nN

an

the sum on the right has only one nonzero term, namely sin mx sin mx dx a .a mm1 11'

8.5 TRIGONOMETRIC SUBSTITUTIONS

1. y 3 tan , , dy , 9 y 9 1 tan ) ) )1 1 ) )) )

)

# ## # "

3 d 9 cos

cos cos 3 39 ycos # #

#

a b k k because cos 0 when ; ) ) 1 1# # 3 ln sec tan C ln C ln 9 y y C' ' 'dy y9 y

cos d d3 cos cos 3 3

9 y

w w ##

#

#

) ) )) )

k k ) ) 2. ; 3y x ; x tan t, t , dx , 1 x ;' '3 dy1 9y

dx dt1 x cos t cos t # # #

"#

#

#

c d 1 1 ln sec t tan t C ln x 1 x C ln 1 9y 3y C' 'dx dt

1 x cos t # ##

#

"

cos t

k k

3. tan tan 1 tan ( 1)' # # # # # #

" " " " "" " "#

#2

2 dx x4 x 4 4 4# 1 1 1

4. tan tan 1 tan 0 0' '0 0

2 2dx dx x8 2x 4 x 4 16 # # # # # # # # #

" " " " " " " "" " "#

!# # 1 1

5. sin sin sin 0 0'0

23 dx x9 x 3 6 6

" " "$#

!"# # 1 1

6. ; t 2x sin t sin sin 0 0' '0 0

1 2 2 1 2 2 1 20

2 dx dt1 4x 1 t 2 4 4

" " ""# #

c d c d 1 1

7. t 5 sin , , dt 5 cos d , 25 t 5 cos ; ) ) ) ) )1 1# # # 25 t dt (5 cos )(5 cos ) d 25 cos d 25 d 25 C' ' ' ' # # # #) ) ) ) ) )1 cos 2 sin 24) ) ) sin cos C sin C sin C 25 25 t t 25 t5 5 5 5

25 t t 25 t# # # #

" " a b ) ) ) # #

8. t sin , , dt cos d , 1 9t cos ; " "# # #3 3) ) ) ) )1 1

1 9t dt (cos )(cos ) d cos d sin cos C sin (3t) 3t 1 9t C' ' ' a b # #" " " "# "3 3 6 6) ) ) ) ) ) ) ) 9. x sec , 0 , dx sec tan d , 4x 49 49 sec 49 7 tan ; 7 7# # # # #) ) ) ) ) ) )

1 sec d ln sec tan C ln C' 'dx 2x

4x 49 sec tan d

7 tan 7 74x 49

#

#

#

" " "# # #

7

) ) )

)' ) ) ) )k k


10. x sec , 0 , dx sec tan d , 25x 9 9 sec 9 3 tan ; 3 35 5) ) ) ) ) ) )1

## #

sec d ln sec tan C ln C' '5 dx 5x25x 9

5 sec tan d3 tan 3 3

25x 9 #

#

35 ) ) )

)' ) ) ) )k k

11. y 7 sec , 0 , dy 7 sec tan d , y 49 7 tan ; ) ) ) ) ) )1# # dy 7 tan d 7 sec 1 d 7(tan ) C' 'y 49y 7 sec (7 tan )(7 sec tan ) d# # # ) ) ) )

)' ') ) ) ) ) )a b

7 sec C y 497 7y# "12. y 5 sec , 0 , dy 5 sec tan d , y 25 5 tan ; ) ) ) ) ) )1# # dy tan cos d sin d (1 cos 2 ) d' ' ' ' 'y 25y 125 sec 5 5 10(5 tan )(5 sec tan ) d#

$ $

" " "# # # ) ) ) ))

) ) ) ) ) ) )

sin cos C sec C C " " " #10 10 5 y y 10 yy y 25 y 255 seca b ) ) ) # #"#

y5

13. x sec , 0 , dx sec tan d , x 1 tan ; ) ) ) ) ) )1# # sin C C' ' 'dx sec tan d d

x x 1 sec tan sec xx 1# #

#

#

) ) ) )) ) )

)

14. x sec , 0 , dx sec tan d , x 1 tan ; ) ) ) ) ) )1# # 2 cos d 2 d sin cos C' ' ' '2 dx 2 tan sec d 1 cos 2

x x 1 sec tan $ # $ #

# ) ) ) )

) )) ) ) ) ) )

tan cos C sec x x 1 C sec x C ) ) )# " "# " # x x

x 1 ##

15. x 2 tan , , dx , x 4 ; ) )1 1 )) )# #

#2 d 2cos cos #

8 8 ;' ' ' 'x dx sin d

x 48 tan (cos ) d cos 1 ( sin ) d

cos cos cos

$ $

#

$ #

# % % a b a b ) ) ) ) ) )

)

) )

) )

t cos 8 dt 8 dt 8 C 8 sec Cc d ) )' 't 1 sect tt t 3t 3# $% %# $ " " " " ) 8 C x 4 4 x 4 C a b a bx 4 x 48 3 3# # $## " # $# #16. x tan , , dx sec d , x 1 sec ; ) ) ) ) )1 1# #

# # C C' ' 'dx sec d cos d

x x 1 tan sec sin sin xx 1# #

#

# #

#

" ) ) ) )) ) ) )

17. w 2 sin , , dw 2 cos d , 4 w 2 cos ; ) ) ) ) )1 1# # # 2 2 cot C C' ' '8 dw 8 2 cos d d

w 4 w 4 sin 2 cos sin w2 4 w# #

# #

#

) ) )

) ) ))

18. w 3 sin , , dw 3 cos d , 9 w 3 cos ; ) ) ) ) )1 1# # # dw cot d d csc 1 d' ' ' ' '9 w

w 9 sin sin3 cos 3 cos d 1 sin # ### # #

#

) ) ) )) )

) ) ) ) ) a b cot C sin C ) ) 9 w

w 3w "#

19. x sin , 0 , dx cos d , 1 x cos ; ) ) ) ) )13 a b# $$# 4 d 4 sec 1 d' ' ' '

0 0 0 0

3 2 3 3 3 4x dx 4 sin cos d 1 cos

1 x cos cos# # #

#

$#

$ #a b # 1 1 1

) ) ) )

) ) a b) ) )

4 tan 4 3 c d ) ) 1 1$! 43


20. x 2 sin , 0 , dx 2 cos d , 4 x 8 cos ; ) ) ) ) )16 a b# $$# tan ' ' '

0 0 0

1 6 6dx 2 cos d d4 x 8 cos 4 cos 4 12

34 3a b

" " "'!#

$#

$ #

1 1

) ) )

) )

1c d)21. x sec , 0 , dx sec tan d , x 1 tan ; ) ) ) ) ) )12 a b# $$# C C' ' 'dx sec tan d cos d x

x 1 tan sin sin x 1a b # $# $ # #"

) ) ) ) )

) ) )

22. x sec , 0 , dx sec tan d , x 1 tan ; ) ) ) ) ) )12 a b# & d C C' ' 'x dx sec sec tan d cos x

x 1 3 x 1tan 3 sinsin# # $

# #

$#

& $%a b a b " ) ) ) ) )

) ))

)

23. x sin , , dx cos d , 1 x cos ; ) ) ) ) )1 1# ## $$#a b

cot csc d C C' ' 'a b 1 x dxx sin 5 5 x

cos cos d cot 1 x % # "

$#

' '

$ &

#

) ) ) ))

) ) ) 24. x sin , , dx cos d , 1 x cos ; ) ) ) ) )1 1# #

# "#a b cot csc d C C' ' 'a b 1 x dx

x sincos cos d cot

3 3 x1 x # # "

$#

"#

% %

$

#

) ) ) ))

) ) ) 25. x tan , , dx sec d , 4x 1 sec ; " "# # # #

# # %#) ) ) ) )

1 1 a b 4 cos d 2( sin cos ) C 2 tan 2x C' ' '8 dx 4x

4x 18 sec d

sec 4x 1a b

a b# #"

#

#

% ## "

) )

)) ) ) ) )

26. t tan , , dt sec d , 9t 1 sec ; " "# ## # #

3 3) ) ) ) )1 1

2 cos d sin cos C tan 3t C' ' '6 dt 3t9t 16 sec d

sec 9t 1a b

a b# #"

#

% ## "

3 ) )

)) ) ) ) )

27. v sin , , dv cos d , 1 v cos ; ) ) ) ) )1 1# ## &a b

tan sec d C C' ' 'v dv sin cos d tan v1 v cos 3 3 1 v# # $

#

&

#a b # # "

$

) ) ) ))

) ) ) 28. r sin , ; ) )1 1# # cot csc d C C' ' 'a b 1 r dr

r sin 7 7 rcos cos d cot 1 r ' # "

(#

) )

& (

#

) ) ) ))

) ) ) 29. Let e 3 tan , t ln (3 tan ), tan tan , dt d , e 9 9 tan 9 3 sec ;t 2t ) ) ) ) ) )" "" %$ $ # sectan # )) sec d ln sec tan ' ' '

0 tan 1 3 tan 1 3

ln 4 tan 4 3 tan 4 3tan 4e dt 3 tan sec d

e 9 tan 3 sec t

2t " "" "

"

) ) )

) )

#

) ) ) )c dk k 3tan 1 3 " ln ln ln 9 ln 1 10 5 43 3 3 310 "

30. Let e tan , t ln (tan ), tan tan , dt d , 1 e 1 tan sec ;t 2t ) ) ) ) ) )" " # # 3 4 sec4 3 tan # )) cos d sin ' ' '

ln 3 4 tan 3 4 tan 3 4

ln 4 3 tan 4 3 tan 4 3tan 4

e dt1 e

(tan ) dsec

t

2ta b

$#$

" "

" "

"

) )

)

sectan #

)

)

) ) )c d $%

3tan"

4 35 5 5

"

31. ; u 2 t, du dt ; u tan , , du sec d , 1 u sec ;' '1 12 1 3

1 4 1

2 dt 2 dut 4t t t 1 u 6 4

"

# # #

#

) ) ) ) )1 1

2 2' '1 3 6

1 4

2 du 2 sec d1 u sec 4 6 6

%'# ##

1

1

) ) 1 1 1

)

1

1c d )


32. y e , 0 , dy e sec d , 1 (ln y) 1 tan sec ; tan tan4) )1) ) ) ) )# # # d sec d ln sec tan ln 1 2' ' '

1

e 4 4dyy 1 (ln y) 0 0

e sece sec

%!#

#

1 1 tan

tan

)

)

)

)

1) ) ) ) )c dk k

33. x sec , 0 , dx sec tan d , x 1 sec 1 tan ; ) ) ) ) ) ) )1# # # C sec x C' 'dx sec tan d

x x 1 sec tan #" ) ) )

) ))

34. x tan , dx sec d , 1 x sec ; ) ) ) )# # #

C tan x C' 'dx sec dx 1 sec# ##

" ) )

))

35. x sec , dx sec tan d , x 1 sec 1 tan ; ) ) ) ) ) ) # # sec d tan C x 1 C' ' 'x dx sec sec tan d

x 1 tan ## # ) ) ) )

)

) ) ) 36. x sin , dx cos d , ; ) ) ) )1 1# # C sin x C' 'dx cos d

1 x cos "#

) ))

)

37. x x 4; dy x 4 ; y dx; x 2 sec , 0dx 2 sec tan d

x 4 2 tan

dydx x x

dx x 4

# # #

#

' #) )

) ) )

)

1

y 2 tan d 2 sec 1 d 2(tan ) C ' ' '(2 tan )(2 sec tan ) d2 sec ) ) ) )) # #) ) ) ) ) )a b 2 sec C; x 2 and y 0 0 0 C C 0 y 2 sec x 4 x 4x x# # # # # #" "

38. x 9 1, dy ; y ; yx 3 sec , 0dx 3 sec tan d

x 9 3 tan

#

#

#

dydx 3 tan

dx dx 3 sec tan dx 9 x 9 # #' '

) )

) ) )

)

1

) ) )

)

sec d ln sec tan C ln C; x 5 and y ln 3 ln 3 ln 3 C C 0 ' ) ) ) )k k x3 3x 9 # y ln x3 3x 9 #

39. x 4 3, dy ; y 3 tan C; x 2 and y 0 0 tan 1 Ca b# " " # # # dydx x 4 x 43 dx dx 3 x 3# #' C y tan 3 3 x 38 8

1 1

# #"

40. x 1 x 1, dy ; x tan , dx sec d , x 1 sec ;a b a b# # # $# $##

dydxdx

x 1a b# $# ) ) ) )

y cos d sin C tan cos C C C; x 0 and y 1 ' 'sec d tan xsec sec x 1#

$

#

) ) )

) )) ) ) ) )

1 0 C y 1 xx 1 #

41. A dx; x 3 sin , 0 , dx 3 cos d , 9 x 9 9 sin 3 cos ; '0

3 9 x3

## ## ) ) ) ) ) )

1 A 3 cos d sin cos ' '

0 0

2 21 1 3 cos 3 cos d 3 33 4

) ) ) 11 ##

#!) ) ) ) )c d


42. V dx 4 ; ' '0 0

1 11 1 2 dx1 x x 1 # # # #a b

x tan , dx sec d , x 1 sec ; ) ) ) )# # #

V 4 4 cos d 1 1 ) )' '0 0

4 41 1 sec d

sec

#

%

) )

)

#

2 (1 cos 2 ) d 2 1 1 ) ) 1 ) 1'0

41 sin 2) 11# #

%

!

43. C C' ' 'dx 2 dz 2 21 sin x (1 z) 1 z1 1 tan 2 dz

1 z2z

1 z

#

#

#

#

x

44. ln 1 z C' ' ' 'dx 2 dz dz1 sin x cos x 1 z 2z 1 z 1 z1

2 dz

1 z2z 1 z

1 z 1 z

#

# #

#

# #

k k ln tan 1 C x#45. (1 2) 1' ' '

0 0 0

2 1 11 dx 2 dz 21 sin x (1 z) 1 z1

"

!

2 dz

1 z2z

1 z

#

#

#

46. 3 1' ' '1

1

3 1 3 1 3

2 1 1dx dz 11 cos x z z1

"

" $

2 dz

1 z1 z1 z

#

#

#

#

47. tan tan ' ' ' '0 0 0 0

2 1 1 11 d 2 dz 2 dz 2 z 22 cos 2 2z 1 z z 32 3 3 3 3

)

)

" ""

!

" 2 dz

1 z1 z1 z

#

#

#

# # #

1 1

3 339

48. ' ' ' '1

1

2 1 1 1

2 3 3 3 3cos d 1

sin cos sin 2z 2z 2z 2z2 1 z dz) )

) ) )

a b1 z 2 dz1 z 1 z2z 1 z

1 z

2z1 z

#

# #

#

#

# #

#

$ $

z2z#

dz

ln z ln 3 0 (ln 3 2) ln 3 1 " " " " " "# # # #$"z 3 ln 34 4 4 4 4#

49. ln C' ' ' 'dt 2 dz 2 dzsin t cos t 2z 1 z (z 1) 2 2 z 1 2

z 1 2

"

2 dz1 z2z 1 z

1 z 1 z

#

# #

#

# #

ln C "

2 tan 1 2

tan 1 2 tt##

50. ' ' ' 'cos t dt1 cos t 1 z 1 z 1 z12 1 z dz 2 1 z dz

1 z 1 z 1 z

a b a ba b a b a b a b a b

1 z 2 dz1 z 1 z

1 z1 z

#

# #

#

#

# #

# # #

# # # #

2 2 tan z C cot t C ' ' ' ' 'a ba b a b1 z dz1 z z z 1 z 1 z z z 1 zdz dz dz dz t #" "#

# # # # # # #

51. sec d' ' ' ' ' ' ') ) d 2 dz 2 dz dz dzcos 1 z (1 z)(1 z) 1 z 1 z)

)

2 dz1 z1 z1 z

#

#

#

#

ln 1 z ln 1 z C ln C k k k k 1 tan1 tan )

)

#

#

52. csc d ln z C ln tan C' ' ' ') ) d dzsin z) )

)

2 dz1 z

2z1 z

#

#

k k #


8.6 INTEGRAL TABLES AND COMPUTER ALGEBRA SYSTEMS

1. tan C' dx 2 x 3x x 3 3 3

" (We used FORMULA 13(a) with a 1, b 3)

2. ln C ln C' dxx x 4 x 4 24 x 4 4

x 4 4 x 4 2

" "

#

(We used FORMULA 13(b) with a 1, b 4)

3. 2 x 2 dx 2 x 2 dx' ' ' ' 'x dx dxx 2 x 2 x 2

(x 2) dx

" "

2 x 2 4 C 2 21 3 1 1 3x 2 x 2 2(x 2) $ "

(We used FORMULA 11 with a 1, b 2, n 1 and a 1, b 2, n 1)

4. ' ' ' ' 'x dx 3 dx dx 3 dx(2x 3) (2x 3) (2x 3)(2x 3) dx

2x 3 2x 3 " "# # # #

$# $# $#

$

2x 3 dx 2x 3 dx C " "# # # # # #

" $ ' ' 3 2 3 22x 3 2x 31 ( 1) " " (2x 3 3) C C "

#

2x 3 2x 3(x 3)


5. x 2x 3 dx (2x 3) 2x 3 dx 2x 3 dx 2x 3 dx 2x 3 dx' ' ' ' ' " "# # # #$ "3 3 C 1 C C " # # # # # 2 3 2 2x 32x 3 2x 35 3 5 5(2x 3) (2x 3) (x 1) & $

$# $#


6. x(7x 5) dx (7x 5)(7x 5) dx (7x 5) dx 7x 5 dx 7x 5 dx' ' ' ' ' $# $# $#" " & $7 7 7 75 5 C 2 C " 7 7 7 7 7 5 49 72 5 27x 5 7x 5 (7x 5) 2(7x 5) ( & C (7x 5)49 714x 4 (We used FORMULA 11 with a 7, b 5, n 5 and a 7, b 5, n 3)

7. dx C' ' 9 4x 9 4xx x ( 4) dxx 9 4x

# #

(We used FORMULA 14 with a 4, b 9) 2 ln C 9 4x 9 4x 9x 9 9 4x 9

"

(We used FORMULA 13(b) with a 4, b 9) ln C

9 4x 9 4x 3x 32 9 4x 3

8. C' 'dx 4 dxx 4x 9 x 4x 9

4x 9( 9)x 18#

(We used FORMULA 15 with a 4, b 9) tan C 4x 99x 9 92 2 4x 99

" (We used FORMULA 13(a) with a 4, b 9) tan C 4x 99x 27 94 4x 9 "

Section 8.6 Integral Tables and Computer Algebra Systems 525

9. x 4x x dx x 2 2x x dx sin C' ' # # # #" (x 2)(2x 3 2) 2 2 x x6 2 x 2 # $ 4 sin C 4 sin C (x 2)(2x 6) 4x x (x 2)(x 3) 4x x6 3

x 2 x 2 " " # #

# #

(We used FORMULA 51 with a 2)

10. dx dx 2 x x sin C x x sin (2x 1) C' ' x xx x

2 x x x " " "# # #

# #" "#"

#

#

"

#

"

#

We used FORMULA 52 with a "#

11. ln C ln C' 'dx dxx 7 x

x 7 x 7 7

7 7 x

x x

7 7 x

" "

#

#

#

#

#

#

We used FORMULA 26 with a 7

12. ln C ln C' 'dx dxx 7 x

x 7 x 7 7

7 7 x

x x

7 7 x

" "

# #

#

#

#

#

We used FORMULA 34 with a 7 13. dx dx 2 x 2 ln C 4 x 2 ln C' ' 4 x 2 x 2 2 x 2 4 x

x x x x # # ## # # # # #

(We used FORMULA 31 with a ) #

14. dx dx x 2 2 sec C x 4 2 sec C' ' x 4 x 2x x

x x# # # # # #" "# #

(We used FORMULA 42 with a ) #

15. 25 p dp 5 p dp 5 p sin C 25 p sin C' ' # # # # # ## # # #" "p p p p5 255 5# (We used FORMULA 29 with a 5)

16. q 25 q dq q 5 q dq sin q 5 q 5 2q C' '# # " # ## # # # #" a b 58 5 8q% sin q 25 q 25 2q C 6258 5 8

q" #" # a b (We used FORMULA 30 with a 5)

17. dr dr sin r 2 r C 2 sin r 4 r C' 'r r 2 r r4 r 2 r# # #

# # #

# # # # #" "" "# # #


18. cosh C ln s s 2 C ln s s 2 C' 'ds ds ss 2

s 2 2 # # #

" # ##

We used FORMULA 36 with a 2 19. tan tan C tan tan C' d 2 5 4 25 4 sin 2 5 4 4 3 3 42 25 16

) 1 ) 1

) # " "

" " )

(We used FORMULA 70 with b 5, c 4, a 2)

20. ln C ln C' d 1 5 4 sin 2 3 cos 24 5 sin 2 4 5 sin 2 6 4 5 sin 22 25 165 4 sin 2 25 16 cos 2) ) )

) ) )

) )

"

(We used FORMULA 71 with a 2, b 4, c 5)


21. e cos 3t dt (2 cos 3t 3 sin 3t) C (2 cos 3t 3 sin 3t) C' 2t e e2 3 132t 2t# #

(We used FORMULA 108 with a 2, b 3)

22. e sin 4t dt ( 3 sin 4t 4 cos 4t) C ( sin 4t 4 cos 4t) C' 3t $ e e( 3) 4 25

# #

3t 3t

(We used FORMULA 107 with a 3, b 4)

23. x cos x dx x cos x dx cos x cos x' ' ' '" " " " " # #" " x x dx x x dx1 1 1 1 1 x 1 x

1 1 1 1

# #

# #

(We used FORMULA 100 with a 1, n 1) cos x sin x x 1 x C cos x sin x x 1 x C x x 4 4# #

# # # # # #" " " "" " " " " "# #


24. x tan x dx x tan x dx tan x tan x' ' ' '" " " " " # # " " " " " a b a bx x dx x x dx1 1 1 1 1 x1 x1 1 1 1 a b# # # # # (We used FORMULA 101 with a 1, n 1) tan x dx (after long division) " x 1 x#

## # " " "'

tan x dx dx tan x x tan x C x tan x x C " x x1 x# #

## # # # # # #" " " # "" " " " " "' ' a ba b

25. ln C' 'ds ds s s 39 s 3 s 2 3 3 s 4 3 s 3a b a b a b " # $ #

# # # # #

3 (We used FORMULA 19 with a 3) ln C s s 318 9 s 108 s 3a b " # 26. ln C' 'd d

22 2 2 2 4 2

22

) ) )

)) )

)

)a b

"

## #

#

#

# #

#

3 We used FORMULA 19 with a 2 ln C )

)

)

)4 2 8 2 22a b

"

#

27. dx' ' 4x 9 4x 9x x 24 dxx 4x 9

#

(We used FORMULA 14 with a 4, b 9) 2 ln C ln C

4x 9 4x 9 9 4x 9 4x 9 3x x 39 4x 9 9

24x 9 3

"

(We used FORMULA 13(b) with a 4, b 9)

28. dx C' ' 9x 4 9x 4x x 29 dxx 9x 4

#

(We used FORMULA 14 with a 9, b 4) tan C tan C 9x 4 9x 4 9x 4x 4 x9 2 9x 4 94

# # #

" " (We used FORMULA 13(a) with a 9, b 4)

29. dt 2 3t 4 ( 4)' ' 3t 4t dtt 3t 4

(We used FORMULA 12 with a 3, b 4) 2 3t 4 4 tan C 2 3t 4 4 tan C 2 3t 4

4 43t 4

" " #

(We used FORMULA 13(a) with a 3, b 4)


30. dt 2 3t 9 9' ' 3t 9t dtt 3t 9

(We used FORMULA 12 with a 3, b 9) 2 3t 9 9 ln C 2 3t 9 3 ln C 19 3t 9 93t 9 9 3t 9 33t 9 3 (We used FORMULA 13(b) with a 3, b 9)

31. x tan x dx tan x dx tan x dx' ' '# " " " " " x x x x2 1 2 1 1 x 3 3 1 x2 1 2 1 $ $# #

(We used FORMULA 101 with a 1, n 2); dx x dx ln 1 x C x tan x dx' ' ' 'x x dx x1 x 1 x$ #

# # # #" # # " a b

tan x ln 1 x C x x3 6 6$ #" #" a b

32. dx x tan x dx tan x dx tan x dx' ' ' 'tan x x x x xx ( 2 1) ( 2 1) 1 x ( 1) 1 x" " "

# # #

# " " " " 2 1 2 1 a b

(We used FORMULA 101 with a 1, n 2); ln x ln 1 x C' ' ' 'x dx dx dx x dx1 x x 1 x x 1 x"

# # # #" # a b k k a b

dx tan x ln x ln 1 x C ' tan xx x

"

#

" "" ##k k a b

33. sin 3x cos 2x dx C' cos 5x cos x10 # (We used FORMULA 62(a) with a 3, b 2)

34. sin 2x cos 3x dx C' cos 5x cos x10 # (We used FORMULA 62(a) with a 2, b 3)

35. 8 sin 4t sin dx sin sin C 8 C' t 8 7t 8 9t7 9 7 9sin sin

# # # 7t 9t# # (We used FORMULA 62(b) with a 4, b ) "#

36. sin sin dt 3 sin sin C' t t t t3 6 6 # (We used FORMULA 62(b) with a , b ) " "3 6

37. cos cos d 6 sin sin C' ) ) ) )3 4 12 7 16 7) # (We used FORMULA 62(c) with a , b ) " "3 4

38. cos cos 7 d sin sin C C' ) ) )2 13 2 15 13 1513 1 15sin sin

) ) " # 13 15) )# # (We used FORMULA 62(c) with a , b 7) "2

39. dx' ' ' ' 'x x 1 x dx dx dxx 1 x 1 x 1x 1 x 1

d x 1$# # #

# # ## #

# " #

a b a b a ba b

ln x 1 tan x C " "# ## "a b x2 1 xa b#

(For the second integral we used FORMULA 17 with a 1)

40. dx 3 3' ' ' ' ' ' 'x 6x dx 6x dx 3 dx dx dxx 3 x 3 x 3 x 3x 3 x 3

d x 3

x 3

#

# # # #

# # # # # ##

#

#

#

#

a b a b a b a b

a b


tan 3 tan C " "" "

a b 3 3 3x 3 x x

x 32 3 3 x 2 3

# # # # $ For the first integral we used FORMULA 16 with a 3; for the third integral we used FORMULA 17 with a 3 tan C " " 2 3 3

x 3 xx 3 2 x 3 a b # #

41. sin x dx; 2 u sin u du 2 sin u duu x

x u

dx 2u du' ' '" " " "#

"

u u1 1 1 1 1 u1 1 1 1

#

u sin u # "

' u du1 u#

#

(We used FORMULA 99 with a 1, n 1) u sin u sin u u 1 u C u sin u u 1 u C # " " # "" " " "# # # ## # (We used FORMULA 33 with a 1) x sin x x x C " "# #" #

42. dx; 2u du 2 cos u du 2 u cos u 1 u Cu x

x u

dx 2u du' ' 'cos x

x

cos uu 1"

"

# " " " #

(We used FORMULA 97 with a 1) 2 x cos x 1 x C "

43. dx; du 2 du 2 sin u u 1 u Cu x

x u

dx 2u du' ' '

x

1 xu 2u u1 u 1 u

#

" "# #

" #

# #

#

sin u u 1 u C " # (We used FORMULA 33 with a 1) sin x x 1 x C sin x x x C " " #

44. dx; 2u du 2 2 u duu x

x u

dx 2u du' ' ' 2 xx

2 uu

# #

#

#

2 2 u sin C u 2 u 2 sin C u u u2 2 2# ## # #" " #

We used FORMULA 29 with a 2 2x x 2 sin C # " #x

45. (cot t) 1 sin t dt ; u sin tdu cos t dt' ' ' # 1 sin t (cos t) dtsin t u1 u du# #

1 u ln C # 1 1 uu

#

(We used FORMULA 31 with a 1) 1 sin t ln C # 1 1 sin t

sin t#


46. ; ln Cu sin tdu cos t dt' ' 'dt cos t dt du

(tan t) 4 sin t (sin t) 4 sin t u 4 u2 4 u

u

"#

# # #

#

(We used FORMULA 34 with a 2) ln C "#

2 4 sin tsin t

#

47. ; ln u 3 u Cu ln yy e

dy e du' ' 'dy

y 3 (ln y)e du du

e 3 u 3 u ## # #

uu

u

u

ln ln y 3 (ln y) C # We used FORMULA 20 with a 3

48. ; ln u 5 u C ln sin 5 sin Cu sin du cos d' 'cos d du

5 sin 5 u) )

) # ## # )) ) ) ) We used FORMULA 20 with a 5

49. ; ln u u 1 C ln 3r 9r 1 Cu 3rdu 3 dr' '3 dr du

9r 1 u 1 # # # # (We used FORMULA 36 with a 1)

50. ; ln u 1 u C ln 3y 1 9y Cu 3ydu 3 dy' '3 dy

1 9ydu

1 u # ## # (We used FORMULA 20 with a 1)

51. cos x dx; 2 t cos t dt 2 cos t dt t cos t dtt xx t

dx 2t dt' ' ' '" " " # "#

# #"

t t t1 t 1 t# # #

# #

(We used FORMULA 100 with a 1, n 1) t cos t sin t t 1 t C # " "" "# # # (We used FORMULA 33 with a 1) x cos x sin x x 1 x C x cos x sin x x x C " " " "" " " "# # # # #

52. tan y dy; 2 t tan t dt 2 tan t dt t tan t dtt yy t

dy 2t dt' ' ' '" " " # "#

# # "

t t t1 t 1 t# # #

# #

(We used FORMULA 101 with n 1, a 1) t tan t dt t tan t t tan t C y tan y tan y y C # " # " " " " ' 't 1 dtt 1 1 t#

# #

53. sin 2x dx sin 2x dx sin 2x dx' ' '& $# # sin 2x cos 2x 5 1 sin 2x cos 2x 4 sin 2x cos 2x 3 15 5 10 5 3 3% % #

(We used FORMULA 60 with a 2, n 5 and a 2, n 3) sin 2x cos 2x cos 2x C C sin 2x cos 2x 2 8 sin 2x cos 2x 2 sin 2x cos 2x 4 cos 2x10 15 15 10 15 15% % ## "

#

54. sin d sin d sin cos sin d' ' '& $ %# # # # # ) ) ) ) )) ) ) sin cos sin cos

5 35 1 2 4 3 1

5 5 5 3% #

# # # #

" "

# #

) ) ) )

We used FORMULA 60 with a , n 5 and a , n 3 " "# # sin cos sin cos 2 cos C sin cos sin cos cos C 2 8 8 2 8 165 15 15 5 15 15

% # % ## # # # # # # # # #) ) ) ) ) ) ) ) ) )


55. 8 cos 2 t dt 8 cos 2 t dt' '% #1 1 cos 2 t sin 2 t 4 14 2 4$ 1 11 (We used FORMULA 61 with a 2 , n 4) 1 6 C cos 2 t sin 2 t t sin (2 2 t)4 2$

1 1

1 1

1 # (We used FORMULA 59 with a 2 ) 1 3t C 3t C cos 2 t sin 2 t 3 sin 4 t cos 2 t sin 2 t 3 cos 2 t sin 2 t4 2$ $

1 1 1 1 1 1 1

1 1 1 1

56. 3 cos 3y dy 3 cos 3y dy' '& $ cos 3y sin 3y5 3 55 1% cos 3y dy cos 3y sin 3y cos 3y sin 3y5 5 3 3 3

12 3 1% #

'

(We used FORMULA 61 with a 3, n 5 and a 3, n 3) cos 3y sin 3y cos 3y sin 3y sin 3y C " % #5 15 15

4 8

57. sin 2 cos 2 d sin 2 cos 2 d' '# $ ## #) ) ) ) ) ) sin 2 cos 2 3 1(2 3) 3 $ #

) )

(We used FORMULA 69 with a 2, m 3, n 2) sin 2 cos 2 d sin 2 d(sin 2 ) C sin 2 cos 2 2 sin 2 cos 2 2 sin 2 cos 2 sin 210 5 10 5 10 15$ # $ # $ # $

) ) ) ) ) ) )' '# #"#) ) ) ) )

58. 9 sin cos d 9 sin cos d' '$ $# $#

) ) ) ) ) ) sin cos 3 13 3 # # #

) ) 3 3 2 sin cos 4 cos sin d # $#) ) ) ) )' We used FORMULA 68 with a 1, n 3, m 3# 2 sin cos 4 cos d(cos ) 2 sin cos 4 cos C # $# # ) ) ) ) ) ) )' 25 2 cos sin C #) ) 4559. 2 sin t sec t dt 2 sin t cos t dt 2 cos t dt' ' '# % # % % sin t cos t 2 12 4 2 4$ (We used FORMULA 68 with a 1, n 2, m 4) sin t cos t cos t dt sin t cos t sec t dt sin t cos t sec t dt $ % $ % $ #

' ' ' sec t tan t 4 24 1 4 1# (We used FORMULA 92 with a 1, n 4) sin t cos t tan t C sec t tan t tan t C tan t sec t 1 C $ # # a bsec t tan t 2 2 2 23 3 3 3 3# tan t C 23

$

An easy way to find the integral using substitution:

2 sin t cos t dt 2 tan t sec t dt 2 tan t d(tan t) tan t C' ' '# % # # # $ 23

60. csc y cos y dy sin y cos y dy sin y cos y dy' ' '# & # & # $ # "sin y cos y

5 2 5 5 1%

sin y cos y dy " "sin y sin ycos y cos y

3 3 3 2 3 4 3 1% # # #'

(We used FORMULA 69 with n 2, m 5, a 1 and n 2, m 3, a 1) cos y sin y d(sin y) C "

sin y cos y3 3 sin y 3 3 sin y 3 sin y 3 sin y

4 8 8cos y 4 cos y%

% # " # #'

61. 4 tan 2x dx 4 tan 2x dx tan 2x 4 tan 2x dx' ' '$ ## tan 2x2# (We used FORMULA 86 with n 3, a 2) tan 2x ln sec 2x C tan 2x 2 ln sec 2x C # ##

4 k k k k


62. tan dx tan dx tan tan dx' ' '% # $ ## # # # x x 2 x xtan(4 1) 3 $ #"#

x

(We used FORMULA 86 with n 4, a ) "# tan 2 tan x C 2 x x3

$# #

(We used FORMULA 84 with a ) "#

63. 8 cot t dt 8 cot t dt' '% # cot t3$ (We used FORMULA 87 with a 1, n 4) 8 cot t cot t t C " $3 (We used FORMULA 85 with a 1)

64. 4 cot 2t dt 4 cot 2t dt cot 2t 4 cot 2t dt' ' '$ # cot 2t2(3 1)#

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