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Problem 8.6.76
Evaluate the following integrals.
∫ x/(x² + 6x + 18) dx
Problem 8.1.44
7–64. Integration review Evaluate the following integrals.
44. ∫ from 0 to √3 of (6x³) / √(x² + 1) dx
Problem 8.9.13
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
13. ∫ (from 0 to ∞) cos x dx
Problem 8.6.9
7–84. Evaluate the following integrals.
9. ∫ from 4 to 6 [1 / √(8x – x²)] 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 8.6.53
7–84. Evaluate the following integrals.
53. ∫ eˣ cot³(eˣ) dx
Problem 8.5.23
23-64. Integration Evaluate the following integrals.
23. ∫ [3 / ((x - 1)(x + 2))] dx
Problem 8.6.22
7–84. Evaluate the following integrals.
22. ∫ [1 / ((x - a)(x - b))] dx, where a ≠ b
Problem 8.4.19
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
19. ∫ 1/√(x² - 81) dx, x > 9
Problem 8.4.48
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
48. ∫ √(9 - 4x²) dx
Problem 8.2.38
9–40. Integration by parts Evaluate the following integrals using integration by parts.
38. ∫ x² ln²(x) dx
Problem 8.9.81
77–86. Comparison Test Determine whether the following integrals converge or diverge.
81. ∫(from 1 to ∞) (sin²x) / x² dx
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 8.6.59
7–84. Evaluate the following integrals.
59. ∫ 1/(x⁴ + x²) dx
Problem 8.9.36
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
36. ∫ (from e² to ∞) 1/(x lnᵖ x) dx, p > 1
Problem 8.2.11
9–40. Integration by parts Evaluate the following integrals using integration by parts.
11. ∫ t · e⁶ᵗ dt
Problem 8.1.7
7–64. Integration review Evaluate the following integrals.
7. ∫ dx / (3 - 5x)^4
Problem 8.5.35
23-64. Integration Evaluate the following integrals.
35. ∫ (x² + 12x - 4)/(x³ - 4x) dx
Problem 8.2.20
9–40. Integration by parts Evaluate the following integrals using integration by parts.
20. ∫ sin⁻¹(x) dx
Problem 8.7.46
41–48. Geometry problems Use a table of integrals to solve the following problems.
46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.
Problem 8.3.5
5. What is a reduction formula?
Problem 8.6.94
92–98. Evaluate the following integrals.
94. ∫ (dt / (t³ + 1))
Problem 8.7.24
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
24. ∫ dt / √(1 + 4eᵗ)
Problem 8.2.23
9–40. Integration by parts Evaluate the following integrals using integration by parts.
23. ∫ x² sin(2x) dx
Problem 8.3.51
9–61. Trigonometric integrals Evaluate the following integrals.
51. ∫ (csc²x + csc⁴x) dx
Problem 8.1.77
Let f(x) = √(x + 1). Find the area of the surface generated when:
Region bounded by f(x) and the x-axis on [0, 1]
Revolved about the x-axis
Problem 8.7.43
41–48. Geometry problems Use a table of integrals to solve the following problems.
43. Find the length of the curve y = eˣ on the interval from 0 to ln 2.
Problem 8.2.72
72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].
Problem 8.9.22
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
22. ∫ (from -∞ to -2) (1/x²) sin(π/2) dx
Problem 8.2.48
48. Integral of sec³x Use integration by parts to show that:
∫ sec³x dx = (1/2) secx tanx + (1/2) ∫ secx dx