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Problem 8.5.26
23-64. Integration Evaluate the following integrals.
26. ∫₀¹ [1 / (t² - 9)] dt
Problem 8.3.65
65. Volume Find the volume of the solid generated when the region bounded by y = sin²(x) * cos^(3/2)(x) and the x-axis on the interval [0, π/2] is revolved about the x-axis.
Problem 8.4.16
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
16. ∫ x²/(25 + x²)² dx
Problem 8.5.23
23-64. Integration Evaluate the following integrals.
23. ∫ [3 / ((x - 1)(x + 2))] dx
Problem 8.2.44
42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.
Problem 8.9.31
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
31. ∫ (from 1 to ∞) 1/[v(v + 1)] dv
Problem 8.1.10
7–64. Integration review Evaluate the following integrals.
10. ∫ e^(3 - 4x) 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 8.2.51
50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:
51. ∫ xⁿ cos(ax) dx = (xⁿ sin(ax))/a - (n/a) ∫ xⁿ⁻¹ sin(ax) dx, for a ≠ 0
Problem 8.3.26
9–61. Trigonometric integrals Evaluate the following integrals.
26. ∫ sin³x cos³ᐟ²x dx
Problem 8.3.37
9–61. Trigonometric integrals Evaluate the following integrals.
37. ∫ [sec⁴(lnθ)]/θ dθ
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 8.3.1
1. State the half-angle identities used to integrate sin²x and cos²x.
Problem 8.4.2
2. What change of variables is suggested by an integral containing √(x² + 36)?
Problem 8.1.1
What change of variables would you use for the integral ∫(4 - 7x)^(-6) dx?
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 8.5.76
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.
76. ∫ [cosθ / (sin³θ - 4sinθ)] dθ
Problem 8.7.52
49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.
52. ∫ from 0 to π/2 of cos⁶x dx
Problem 8.4.71
{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,
∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C
Graph the following functions and find the area under the curve on the given interval.
f(x) = (9 - x²) ⁻², [0, 3/2]
Problem 8.1.32
7–64. Integration review Evaluate the following integrals.
32. ∫ from 0 to 2 of x / (x² + 4x + 8) dx
Problem 8.9.106
102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:
F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).
Verify the following Laplace transforms, where a is a real number.
106. f(t) = cos(at) → F(s) = s/(s² + a²)
Problem 8.7.37
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.
37. ∫ dx / √(x² + 10x), x >
Problem 8.9.56
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
56. ∫ (from 0 to 1) 1/(x + √x) dx
Problem 8.9.28
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
28. ∫ (from 1 to ∞) tan⁻¹(s)/(s² + 1) ds
Problem 8.7.21
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.
21. ∫ cos x / (sin² x + 2 sin x) dx
Problem 8.1.40
7–64. Integration review Evaluate the following integrals.
40. ∫ (1 - x) / (1 - √x) dx
Problem 8.8.38
37-40. {Use of Tech} Temperature data
Howdy temperature data for Boulder, Colorado; San Francisco, California; Nantucket, Massachusetts; and Duluth, Minnesota, over a 12-hr period on the same day of January are shown in the figure.
Assume these data are taken from a continuous temperature function T(t). The average temperature (in °F) over the 12-hr period is:
T_avg = (1/12) × ∫(0 to 12) T(t) dt
38. Find an accurate approximation to the average temperature over the 12-hr period for San Francisco. State your method.
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 8.3.45
9–61. Trigonometric integrals Evaluate the following integrals.
45. ∫ sec²x tan¹ᐟ²x dx
Problem 8.6.74
Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx
