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Problem 8.7.42
41–48. Geometry problems Use a table of integrals to solve the following problems.
42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.
Problem 8.2.23
9–40. Integration by parts Evaluate the following integrals using integration by parts.
23. ∫ x² sin(2x) dx
Problem 8.5.79
76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
79. ∫ [sec t / (1 + sin t)] dt
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
Problem 8.3.16
9–61. Trigonometric integrals Evaluate the following integrals.
16. ∫ sin²θ cos⁵θ dθ
Problem 8.6.76
Evaluate the following integrals.
∫ x/(x² + 6x + 18) dx
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 8.1.53
7–64. Integration review Evaluate the following integrals.
53. ∫ eˣ sec(eˣ + 1) dx
Problem 8.8.27
27. {Use of Tech} Midpoint Rule, Trapezoid Rule, and relative error
Find the Midpoint and Trapezoid Rule approximations to ∫(0 to 1) sin(πx) dx using n = 25 subintervals. Compute the relative error of each approximation.
Problem 8.6.9
7–84. Evaluate the following integrals.
9. ∫ from 4 to 6 [1 / √(8x – x²)] dx
Problem 8.5.47
23-64. Integration Evaluate the following integrals.
47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx
Problem 8.1.51
7–64. Integration review Evaluate the following integrals.
51. ∫ from -1 to 0 of x / (x² + 2x + 2) dx
Problem 8.9.96
95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.
96. ∫(from 0 to ∞) (sin²x)/x² dx = π/2
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 8.4.51
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
51. ∫ x²/√(4 + x²) dx
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 8.4.33
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
33. ∫ √(x² - 9)/x dx, x > 3
Problem 8.6.49
7–84. Evaluate the following integrals.
49. ∫ tan³x · sec⁹x dx
Problem 8.3.34
9–61. Trigonometric integrals Evaluate the following integrals.
34. ∫ tan⁹x sec⁴x dx
Problem 8.4.65
60–69. Completing the square Evaluate the following integrals.
65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)
Problem 8.9.44
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
44. ∫ (from 0 to ln 3) eʸ/(eʸ-1)⁷ᐟ³ dy
Problem 8.7.18
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.
18. ∫ dx / (225 − 16x²)
Problem 8.4.48
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
48. ∫ √(9 - 4x²) dx
Problem 8.6.19
7–84. Evaluate the following integrals.
19. ∫ from 0 to π/2 [sin⁷x] dx
Problem 8.6.27
7–84. Evaluate the following integrals.
27. ∫ sin⁴(x/2) dx
Problem 8.7.15
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.
15. ∫ x / √(4x + 1) dx
Problem 8.2.14
9–40. Integration by parts Evaluate the following integrals using integration by parts.
14. ∫ s · e⁻²ˢ ds
Problem 8.9.81
77–86. Comparison Test Determine whether the following integrals converge or diverge.
81. ∫(from 1 to ∞) (sin²x) / x² dx
Problem 8.3.51
9–61. Trigonometric integrals Evaluate the following integrals.
51. ∫ (csc²x + csc⁴x) dx
Problem 8.3.45
9–61. Trigonometric integrals Evaluate the following integrals.
45. ∫ sec²x tan¹ᐟ²x dx