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

Section 7.2 Trigo no metric Integrals 485
31. y tan 5 x dx 32. y tan 2 x sec x dx
33. y x sec x tan x dx 34. y sin
cos 3 d
35. y y2
37. y y2
y6 cot2 x dx
y4 cot5 csc 3 d
36. y y2
y4 cot3 x dx
38. y y2
csc 4 cot 4 d
y4
39. y csc x dx 40. y y3
y6 csc3 x dx
41. y sin 8x cos 5x dx 42. y sin 2 sin 6 d
; 59–60 Use a graph of the integrand to guess the value of the
integral. Then use the methods of this section to prove that your
guess is correct.
59. y 2
cos 3 x dx 60. y 2
sin 2x cos 5x dx
0
61–64 Find the volume obtained by rotating the region bounded
by the curves about the given axis.
61. y − sin x, y − 0, y2 < x < ; about the x-axis
62. y − sin 2 x, y − 0, 0 < x < ; about the x-axis
63. y − sin x, y − cos x, 0 < x < y4; about y − 1
64. y − sec x, y − cos x, 0 < x < y3; about y − 21
0
43. y y2
cos 5t cos 10t dt
0
45. y y6
s1 1 cos 2x dx
0
47. y 1 2 tan2 x
sec 2 x
49. y x tan 2 x dx
dx
44. y sin x sec 5 x dx
46. y y4
s1 2 cos 4 d
48. y
0
dx
cos x 2 1
65. A particle moves on a straight line with velocity function
vstd − sin t cos 2 t. Find its position function s − f std if
f s0d − 0.
66. Household electricity is supplied in the form of alternating
current that varies from 155 V to 2155 V with a frequency
of 60 cycles per second (Hz). The voltage is thus given by
the equation
Estd − 155 sins120td
;
50. If y y4
0
tan 6 x sec x dx − I, express the value of
y y4
0
tan 8 x sec x dx in terms of I.
51–54 Evaluate the indefinite integral. Illustrate, and check that
your answer is reasonable, by graphing both the integrand and its
antiderivative (taking C − 0d.
51. y x sin 2 sx 2 d dx 52. y sin 5 x cos 3 x dx
53. y sin 3x sin 6x dx 54. y sec 4 ( 1 2 x) dx
55. Find the average value of the function f sxd − sin 2 x cos 3 x on
the interval f2, g.
56. Evaluate y sin x cos x dx by four methods:
(a) the substitution u − cos x
(b) the substitution u − sin x
(c) the identity sin 2x − 2 sin x cos x
(d) integration by parts
Explain the different appearances of the answers.
57–58 Find the area of the region bounded by the given curves.
57. y − sin 2 x, y − sin 3 x, 0 < x <
58. y − tan x, y − tan 2 x, 0 < x < y4
where t is the time in seconds. Voltmeters read the RMS
(root-mean-square) voltage, which is the square root of the
average value of fEstdg 2 over one cycle.
(a) Calculate the RMS voltage of household current.
(b) Many electric stoves require an RMS voltage of
220 V. Find the corresponding amplitude A needed
for the voltage Estd − A sins120td.
67–69 Prove the formula, where m and n are positive integers.
67. y sin mx cos nx dx − 0
2
68. y sin mx sin nx dx −H 0 2
69. y cos mx cos nx dx −H 0 2
if m ± n
if m − n
if m ± n
if m − n
70. A finite Fourier series is given by the sum
f sxd − o N
n−1
a n sin nx
− a 1 sin x 1 a 2 sin 2x 1 ∙ ∙ ∙ 1 a N sin Nx
Show that the mth coefficient a m is given by the formula
a m − 1 y f sxd sin mx dx
2
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