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Problem 3.3.32
9–61. Trigonometric integrals Evaluate the following integrals.
32. ∫ cot⁵(3x) dx
Problem 5.4.2
If ƒ is an even function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 2 ∫₀ᵃ ƒ(𝓍) d𝓍?
Problem 8.1.44
7–64. Integration review Evaluate the following integrals.
44. ∫ from 0 to √3 of (6x³) / √(x² + 1) dx
Problem 8.3.51
9–61. Trigonometric integrals Evaluate the following integrals.
51. ∫ (csc²x + csc⁴x) dx
Problem 8.4.48
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
48. ∫ √(9 - 4x²) dx
Problem 8.2.1
1. On which derivative rule is integration by parts based?
Problem 8.5.15
5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
15. x / ((x⁴ - 16)²)
Problem 8.5.66
66-68. Areas of regions (Use of Tech) Find the area of the following regions.
66. The region bounded by the curve y = (x - x²)/[(x + 1)(x² + 1)] and the x-axis from x = 0 to x = 1
Problem 8.1.53
7–64. Integration review Evaluate the following integrals.
53. ∫ eˣ sec(eˣ + 1) dx
Problem 8.1.4
Let f(x) = (4x³ + x² + 4x + 2) / (x² + 1). Use long division to show that f(x) = 4x + 1 + 1 / (x² + 1) and use this result to evaluate ∫f(x) dx.
Problem 8.6.59
7–84. Evaluate the following integrals.
59. ∫ 1/(x⁴ + x²) dx
Problem 8.6.94
92–98. Evaluate the following integrals.
94. ∫ (dt / (t³ + 1))
Problem 8.6.76
Evaluate the following integrals.
∫ x/(x² + 6x + 18) dx
Problem 8.5.23
23-64. Integration Evaluate the following integrals.
23. ∫ [3 / ((x - 1)(x + 2))] dx
Problem 8.5.9
5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
9. 4/(x⁵ - 5x³ + 4x)
Problem 8.1.1
What change of variables would you use for the integral ∫(4 - 7x)^(-6) dx?
Problem 8.5.44
23-64. Integration Evaluate the following integrals.
44. ∫₁² 2/[t³(t + 1)] dt
Problem 8.4.30
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
30. ∫ x³√(1 - x²) dx
Problem 8.3.34
9–61. Trigonometric integrals Evaluate the following integrals.
34. ∫ tan⁹x sec⁴x dx
Problem 8.4.19
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
19. ∫ 1/√(x² - 81) dx, x > 9
Problem 8.1.24
7–64. Integration review Evaluate the following integrals.
24. ∫ from 0 to θ of (x⁵⸍² - x¹⸍²) / x³⸍² dx
Problem 8.5.35
23-64. Integration Evaluate the following integrals.
35. ∫ (x² + 12x - 4)/(x³ - 4x) dx
Problem 8.1.7
7–64. Integration review Evaluate the following integrals.
7. ∫ dx / (3 - 5x)^4
Problem 8.1.51
7–64. Integration review Evaluate the following integrals.
51. ∫ from -1 to 0 of x / (x² + 2x + 2) dx
Problem 8.8.54
54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
54. ∫(from 0 to π/2) sin⁶x dx = 5π/32
Problem 8.4.59
59. Area of a segment of a circle
Use two approaches to show that the area of a cap (or segment) of a circle of radius r subtended by an angle θ (see figure) is given by:
A_seg = (1/2) r² (θ - sin θ)
b. Find the area using calculus.
Problem 8.8.18
15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.
18. ∫(0 to 1) e⁻ˣ dx using n = 8 subintervals
Problem 8.7.1
1. Give some examples of analytical methods for evaluating integrals.
Problem 8.2.38
9–40. Integration by parts Evaluate the following integrals using integration by parts.
38. ∫ x² ln²(x) dx
Problem 8.6.30
7–84. Evaluate the following integrals.
30. ∫ from 5/2 to 5√3/2 [1 / (v² √(25 - v²))] dv
