Calculus 2nd Edition Rogawski

SECTION 8.2 Trigonometric Integrals 425
In Exercises 15–18, evaluate the integral using the method described
on page 422 and the reduction formulas on page 423 as necessary.
∫
∫
15. tan 3 x sec xdx 16. tan 2 x sec xdx
∫
17. tan 2 x sec 4 xdx 18.
∫
tan 8 x sec 2 xdx
In Exercises 19–22, evaluate using methods similar to those that apply
to integral tan m x sec n .
∫
∫
19. cot 3 xdx 20. sec 3 xdx
∫
21. cot 5 x csc 2 xdx 22.
∫
cot 4 x csc xdx
In Exercises 23–46, evaluate the integral.
∫
∫
23. cos 5 x sin xdx 24. cos 3 (2 − x)sin(2 − x)dx
∫
∫
25. cos 4 (3x + 2)dx 26. cos 7 3xdx
∫
∫
27. cos 3 (πθ) sin 4 (πθ)dθ 28. cos 498 y sin 3 ydy
∫
∫
29. sin 4 (3x)dx 30. sin 2 x cos 6 xdx
31.
∫
∫
csc 2 (3 − 2x)dx 32. csc 3 xdx
∫
∫
33. tan x sec 2 xdx 34. tan 3 θ sec 3 θ dθ
∫
∫
35. tan 5 x sec 4 xdx 36. tan 4 x sec xdx
∫
∫
37. tan 6 x sec 4 xdx 38. tan 2 x sec 3 xdx
∫
∫
39. cot 5 x csc 5 xdx 40. cot 2 x csc 4 xdx
∫
∫
41. sin 2x cos 2xdx 42. cos 4x cos 6xdx
∫
∫
43. t cos 3 (t 2 tan 3 (ln t)
)dt 44.
dt
t
∫
∫
45. cos 2 (sin t)cos tdt 46. e x tan 2 (e x )dx
In Exercises 47–60, evaluate the definite integral.
∫ 2π
∫ π/2
47. sin 2 xdx 48. cos 3 xdx
0
0
∫ π/2
∫ π/2
49. sin 5 xdx 50. sin 2 x cos 3 xdx
0
0
∫ π/4
∫
dx
π/2 dx
51.
52.
0 cos x
π/4 sin x
∫ π/3
∫ π/4
53. tan xdx 54. tan 5 xdx
0
0
∫ π/4
∫ 3π/2
55. sec 4 xdx 56. cot 4 x csc 2 xdx
−π/4
π/4
∫ π
∫ π
57. sin 3x cos 4xdx 58. sin x sin 3xdx
0
0
∫ π/6
∫ π/4
59. sin 2x cos 4xdx 60. sin 7x cos 2xdx
0
0
61. Use the identities for sin 2x and cos 2x on page 420 to verify that
the following formulas are equivalent.
∫
sin 4 xdx= 1 (12x − 8 sin 2x + sin 4x) + C
32
∫
sin 4 xdx= − 1 4 sin3 x cos x − 3 8 sin x cos x + 3 8 x + C
62. Evaluate ∫ sin 2 x cos 3 xdxusing the method described in the text
and verify that your result is equivalent to the following result produced
by a computer algebra system.
∫
sin 2 x cos 3 xdx= 1 30 (7 + 3 cos 2x)sin3 x + C
63. Find the volume of the solid obtained by revolving y = sin x for
0 ≤ x ≤ π about the x-axis.
64. Use Integration by Parts to prove Eqs. (1) and (2).
In Exercises 65–68, use the following alternative method for evaluating
the integral J = ∫ sin m x cos n xdxwhen m and n are both even. Use
the identities
sin 2 x = 1 2 (1 − cos 2x), cos2 x = 1 (1 + cos 2x)
2
to write J = 1 ∫
4 (1 − cos 2x) m/2 (1 + cos 2x) n/2 dx, and expand the
right-hand side as a sum of integrals involving smaller powers of sine
and cosine in the variable 2x.
∫
∫
65. sin 2 x cos 2 xdx 66. cos 4 xdx
∫
∫
67. sin 4 x cos 2 xdx 68. sin 6 xdx
69. Prove the reduction formula
∫
tan k xdx= tank−1 ∫
x
−
k − 1
tan k−2 xdx
Hint: tan k x = (sec 2 x − 1) tan k−2 x.
∫
70. Use the substitution u = csc x − cot x to evaluate csc xdx(see
Example 5).
∫ π/2
71. Let I m = sin m xdx.
0
(a) Show that I 0 = π 2 and I 1 = 1.
(b) Prove that, for m ≥ 2,
I m = m − 1
m
I m−2
(c) Use (a) and (b) to compute I m for m = 2, 3, 4, 5.
∫ π
72. Evaluate sin 2 mx dx for m an arbitrary integer.
∫ 0
73. Evaluate sin x ln(sin x)dx. Hint: Use Integration by Parts as a
first step.