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Problem 8.6.9
7–84. Evaluate the following integrals.
9. ∫ from 4 to 6 [1 / √(8x – x²)] dx
Problem 8.6.13
7–84. Evaluate the following integrals.
13. ∫ [1 / (eˣ √(1 – e²ˣ))] dx
Problem 8.6.19
7–84. Evaluate the following integrals.
19. ∫ from 0 to π/2 [sin⁷x] dx
Problem 8.6.22
7–84. Evaluate the following integrals.
22. ∫ [1 / ((x - a)(x - b))] dx, where a ≠ b
Problem 8.6.25
7–84. Evaluate the following integrals.
25. ∫ [1 / (x√(1 - x²))] dx
Problem 8.6.27
7–84. Evaluate the following integrals.
27. ∫ sin⁴(x/2) dx
Problem 8.6.30
7–84. Evaluate the following integrals.
30. ∫ from 5/2 to 5√3/2 [1 / (v² √(25 - v²))] dv
Problem 8.6.79
7–84. Evaluate the following integrals.
79. ∫ (arcsinx)/x² dx
Problem 8.6.82
7–84. Evaluate the following integrals.
82. ∫ 1/(1 + tanx) dx
Problem 8.6.92
92–98. Evaluate the following integrals.
92. ∫[1 to √2] y⁸ e^(y²) dy
Problem 8.6.94
92–98. Evaluate the following integrals.
94. ∫ (dt / (t³ + 1))
Problem 8.6.97
92–98. Evaluate the following integrals.
97. ∫ tan⁻¹(∛x) dx
Problem 8.6.2
Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.
∫ (1 + tan x) sec²x dx
Problem 8.6.5
Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.
∫ (5x² + 18x + 20) / [(2x + 3)(x² + 4x + 8)] dx
Problem 8.6.33
7–84. Evaluate the following integrals.
33. ∫ [eˣ / (a² + e²ˣ)] dx, where a ≠ 0
Problem 8.6.35
7–84. Evaluate the following integrals.
35. ∫ from 0 to π/4 [(tan²θ + tanθ + 1) sec²θ] dθ
Problem 8.6.38
7–84. Evaluate the following integrals.
38. ∫ from π/6 to π/2 [cos x · ln(sin x)] dx
Problem 8.6.41
7–84. Evaluate the following integrals.
41. ∫ cot^(3/2)x · csc⁴x dx
Problem 8.6.47
7–84. Evaluate the following integrals.
47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx
Problem 8.6.49
7–84. Evaluate the following integrals.
49. ∫ tan³x · sec⁹x dx
Problem 8.6.53
7–84. Evaluate the following integrals.
53. ∫ eˣ cot³(eˣ) dx
Problem 8.6.56
7–84. Evaluate the following integrals.
56. ∫ from π to 3π/2 sin2x e^(sin²x) dx
Problem 8.6.59
7–84. Evaluate the following integrals.
59. ∫ 1/(x⁴ + x²) dx
Problem 8.6.62
7–84. Evaluate the following integrals.
62. ∫ from 0 to π/2 √(1 + cosθ) dθ
Problem 8.6.65
Evaluate the following integrals.
65. ∫ from 0 to 1/6 1/√(1 - 9x²) dx
Problem 8.6.68
Evaluate the following integrals.
∫ eˣ/(e²ˣ + 2eˣ + 17) dx
Problem 8.6.71
Evaluate the following integrals.
∫ (1 - cosx)/(1 + cosx) dx
Problem 8.6.74
Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx
Problem 8.6.76
Evaluate the following integrals.
∫ x/(x² + 6x + 18) dx
Problem 8.6.85a
85. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. More than one integration method can be used to evaluate ∫ (1 / (1 - x²)) dx.