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

Section 7.1 Integration by Parts 477
31. y 5
1
M
e dM 32. y 2 sln xd 2
dx
M 1 x 3
33. y y3
sin x lnscos xd dx 34. y 1 r 3
dr
0
0
s4 1 r 2
35. y 2
x 4 sln xd 2 dx
1
36. y t
e s sinst 2 sd ds
0
50. Prove that, for even powers of sine,
y y2
sin 2n x dx − 1 ? 3 ? 5 ? ∙ ∙ ∙ ? s2n 2 1d
0
2 ? 4 ? 6 ? ∙ ∙ ∙ ? 2n
51–54 Use integration by parts to prove the reduction formula.
51. y sln xd n dx − xsln xd n 2 n y sln xd n21 dx
2
37–42 First make a substitution and then use integration by
parts to evaluate the integral.
37. y e sx dx 38. y cossln xd dx
39. y s
sy2
3 coss 2 d d
40. y
e cos t sin 2t dt
0
52. y x n e x dx − x n e x 2 n y x n21 e x dx
53. y tan n x dx − tann21 x
n 2 1 2 y tan n22 x dx sn ± 1d
54. y sec n x dx − tan x secn22 x
n 2 1
1 n 2 2
n 2 1 y sec n22 x dx sn ± 1d
41. y x lns1 1 xd dx 42. y
arcsinsln xd
x
dx
55. Use Exercise 51 to find y sln xd 3 dx.
56. Use Exercise 52 to find y x 4 e x dx.
;
43–46 Evaluate the indefinite integral. Illustrate, and check
that your answer is reasonable, by graphing both the function
and its antiderivative (take C − 0).
43. y xe 22x dx 44. y x 3y2 ln x dx
45. y x 3 s1 1 x 2 dx 46. y x 2 sin 2x dx
47. (a) Use the reduction formula in Example 6 to show that
y sin 2 x dx − x 2
2
sin 2x
4
1 C
(b) Use part (a) and the reduction formula to evaluate
y sin 4 x dx.
48. (a) Prove the reduction formula
y cos n x dx − 1 n cosn21 x sin x 1 n 2 1
n
y cos n22 x dx
(b) Use part (a) to evaluate y cos 2 x dx.
(c) Use parts (a) and (b) to evaluate y cos 4 x dx.
49. (a) Use the reduction formula in Example 6 to show that
y y2
sin n x dx − n 2 1
0
n
y y2
sin n22 x dx
0
where n > 2 is an integer.
(b) Use part (a) to evaluate y y2
sin 3 x dx and y y2
0 0
sin 5 x dx.
(c) Use part (a) to show that, for odd powers of sine,
y y2
sin 2n11 x dx −
0
2 ? 4 ? 6 ? ∙ ∙ ∙ ? 2n
3 ? 5 ? 7 ? ∙ ∙ ∙ ? s2n 1 1d
;
57–58 Find the area of the region bounded by the given curves.
57. y − x 2 ln x, y − 4 ln x 58. y − x 2 e 2x , y − xe 2x
59–60 Use a graph to find approximate x-coordinates of the
points of intersection of the given curves. Then find (approximately)
the area of the region bounded by the curves.
59. y − arcsins 1 2 xd, y − 2 2 x 2
60. y − x lnsx 1 1d, y − 3x 2 x 2
61–64 Use the method of cylindrical shells to find the volume
generated by rotating the region bounded by the curves about the
given axis.
61. y − cossxy2d, y − 0, 0 < x < 1; about the y-axis
62. y − e x , y − e 2x , x − 1; about the y-axis
63. y − e 2x , y − 0, x − 21, x − 0; about x − 1
64. y − e x , x − 0, y − 3; about the x-axis
65. Calculate the volume generated by rotating the region
bounded by the curves y − ln x, y − 0, and x − 2 about
each axis.
(a) The y-axis
(b) The x-axis
66. Calculate the average value of f sxd − x sec 2 x on the
interval f0, y4g.
67. The Fresnel function Ssxd − y x 0 sin(1 2 t 2 ) dt was discussed
in Example 5.3.3 and is used extensively in the theory of
optics. Find y Ssxd dx. [Your answer will involve Ssxd.]
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.