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Problem 8.2.55
54-57. Applying Reduction Formulas Use the reduction formulas from Exercises 50-53 to evaluate the following integrals:
55. ∫ x² cos(5x) dx
Problem 8.2.20
9–40. Integration by parts Evaluate the following integrals using integration by parts.
20. ∫ sin⁻¹(x) dx
Problem 8.5.32
23-64. Integration Evaluate the following integrals.
32. ∫ (4x - 2)/(x³ - x) dx
Problem 8.1.20
7–64. Integration review Evaluate the following integrals.
20. ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) dx
Problem 8.3.47
9–61. Trigonometric integrals Evaluate the following integrals.
47. ∫ (csc⁴x)/(cot²x) dx
Problem 8.4.23
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
23. ∫ 1/(25 - x²)^(3/2) dx
Problem 8.6.94
92–98. Evaluate the following integrals.
94. ∫ (dt / (t³ + 1))
Problem 8.7.74
71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
74. ∫xⁿ arcsin(x) dx (Hint: integration by parts.)
Problem 8.2.7
Use a substitution to reduce the following integrals to ∫ ln u du. Then evaluate using the formula for ∫ ln x dx.
7. ∫ (sec²x) · ln(tan x + 2) dx
Problem 8.1.68
68. Different methods
a. Evaluate ∫(cot x csc² x) dx using the substitution u=cotx.
Problem 8.4.74
{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) = 1/(x√(x² - 36)), [12/√3 , 12]
Problem 8.3.7
7. How would you evaluate ∫ tan¹⁰x sec²x dx?
Problem 8.9.4
4. Evaluate ∫ (from 0 to 1) (1/x^(1/5)) dx after writing the integral as a limit.
Problem 8.6.56
7–84. Evaluate the following integrals.
56. ∫ from π to 3π/2 sin2x e^(sin²x) dx
Problem 8.1.40
7–64. Integration review Evaluate the following integrals.
40. ∫ (1 - x) / (1 - √x) dx
Problem 8.9.98
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.
98. ∫(from 0 to 1) (ln x)/(1+x) dx = -π²/12
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 8.3.4
4. Describe the method used to integrate sinᵐx cosⁿx, for m even and n odd.
Problem 8.3.60
9–61. Trigonometric integrals Evaluate the following integrals.
60. ∫ from 0 to π/8 of √(1 - cos8x) dx
Problem 8.7.31
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.
31. ∫ √(x² - 8x) dx, x > 8
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 8.9.19
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
19. ∫ (from 1 to ∞) (3x² + 1)/(x³ + x) dx
Problem 8.7.76
76. Apparent discrepancy
Three different computer algebra systems give the following results:
∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).
Explain how all three can be correct.
Problem 8.4.26
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x dx
Problem 8.7.71
71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
71. ∫[x/(ax + b)] dx (Hint: u = ax + b.)
Problem 8.1.60
7–64. Integration review Evaluate the following integrals.
60. ∫ from 0 to π/4 of 3√(1 + sin 2x) dx
Problem 8.6.25
7–84. Evaluate the following integrals.
25. ∫ [1 / (x√(1 - x²))] dx
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 8.8.63
63. (Use of Tech) Normal distribution of heights
The heights of U.S. men are normally distributed with a mean of 69 in and a standard deviation of 3 in. This means that the fraction of men with a height between a and b (with a < b) inches is given by the integral
(1/(3√(2π))) ∫ₐᵇ e^(-((x-69)/3)²/2) dx.
What percentage of American men are between 66 and 72 inches tall? Use the method of your choice, and experiment with the number of subintervals until you obtain successive approximations that differ by less than 10⁻³.
Problem 8.3.45
9–61. Trigonometric integrals Evaluate the following integrals.
45. ∫ sec²x tan¹ᐟ²x dx
