James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 5.5 The Substitution Rule 419
;
13. y
dx
5 2 3x
14. y y 2 s4 2 y 3 d 2y3 dy
15. y cos 3 sin d 16. y e 25r dr
17. y
19. y
21. y
e u
s1 2 e u d du 2
a 1 bx 2
dx
s3ax 1 bx 3
sln xd2
dx
x
18. sin sx
y dx
sx
z
20. y
2
z 3 1 1 dz
22. y sin x sinscos xd dx
23. y sec 2 tan 3 d 24. y x sx 1 2 dx
25. y e x s1 1 e x dx 26. y
dx
ax 1 b
27. y sx 2 1 1dsx 3 1 3xd 4 dx 28. y e cos t sin t dt
29. y 5 t sins5 t d dt 30. y sec 2 x
tan 2 x dx
31. y
sarctan xd2
x 2 1 1
dx
32. y
x
x 2 1 4 dx
sa ± 0d
33. y cos s1 1 5td dt 34. y cossyxd
x 2 dx
35. y scot x csc 2 x dx 36. y
37. y sinh 2 x cosh x dx 38. y
39. y
sin 2x
1 1 cos 2 x dx 40. y
41. y cot x dx 42. y
dx
43. y
s1 2 x 2 sin 21 x
44. y
2 t
2 t 1 3 dt
dt
cos 2 ts1 1 tan t
sin x
1 1 cos 2 x dx
cossln td
dt
t
x
1 1 x dx 4
45. y 1 1 x
1 1 x 2 dx 46. y x 2 s2 1 x dx
47. y xs2x 1 5d 8 dx 48. y x 3 sx 2 1 1 dx
49–52 Evaluate the indefinite integral. Illustrate and check that
your answer is reasonable by graphing both the function and its
antiderivative (take C − 0).
49. y xsx 2 2 1d 3 dx 50. y tan 2 sec 2 d
51. y e cos x sin x dx 52. y sin x cos 4 x dx
53–73 Evaluate the definite integral.
53. y 1
cossty2d dt 54. y 1
s3t 2 1d 50 dt
0
0
;
55. y 1
1 1 7x dx 56. y 3 dx
0 s3 0 5x 1 1
57. y y6
0
59. y 2
1
61. y y4
63. y 13
sin t
cos 2 t dt
e 1yx
58. y 2y3
y3
csc 2 ( 1 2t) dt
x dx 60. y 1
xe 2x 2 dx
2 0
2y4 sx 3 1 x 4 tan xd dx
0
dx
s 3 s1 1 2xd 2
65. y a
xsx 2 1 a 2 dx sa . 0d 66. y y3
0
62. y y2
cos x sinssin xd dx
0
64. y a
xsa 2 2 x 2 dx
0
2y3 x 4 sin x dx
67. y 2
xsx 2 1 dx 68. y 4 x
dx
1
0
s1 1 2x
dx
69. y e4
70. y 2
sx 2 1de sx21d 2 dx
e xsln x 0
71. y 1
0
73. y 1
0
e z 1 1
e z 1 z dz
dx
(1 1 sx ) 4
72. y Ty2
sins2tyT 2 d dt
74. Verify that f sxd − sin s 3 x is an odd function and use that fact
to show that
0 < y 3 sin s 3 x dx < 1
22
75–76 Use a graph to give a rough estimate of the area of the
region that lies under the given curve. Then find the exact area.
75. y − s2x 1 1, 0 < x < 1
76. y − 2 sin x 2 sin 2x, 0 < x <
77. Evaluate y 2 22 sx 1 3ds4 2 x 2 dx by writing it as a sum of
two integrals and interpreting one of those integrals in terms
of an area.
78. Evaluate y 1 0 xs1 2 x 4 dx by making a substitution and
interpreting the resulting integral in terms of an area.
79. Which of the following areas are equal? Why?
y
y=e œ„x
0 1 x
y
y=e sin x sin 2x
y
0
y=2x´
0 1 x
0 1 π x
2
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