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Problem 8.3.51
9–61. Trigonometric integrals Evaluate the following integrals.
51. ∫ (csc²x + csc⁴x) dx
Problem 8.5.26
23-64. Integration Evaluate the following integrals.
26. ∫₀¹ [1 / (t² - 9)] dt
Problem 8.1.57
7–64. Integration review Evaluate the following integrals.
57. ∫ dx / (x¹⸍² + x³⸍²)
Problem 8.5.32
23-64. Integration Evaluate the following integrals.
32. ∫ (4x - 2)/(x³ - x) dx
Problem 8.6.74
Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx
Problem 8.5.50
23-64. Integration Evaluate the following integrals.
50. ∫ 8(x² + 4)/[x(x² + 8)] dx
Problem 8.3.16
9–61. Trigonometric integrals Evaluate the following integrals.
16. ∫ sin²θ cos⁵θ dθ
Problem 8.6.2
Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.
∫ (1 + tan x) sec²x dx
Problem 8.2.23
9–40. Integration by parts Evaluate the following integrals using integration by parts.
23. ∫ x² sin(2x) dx
Problem 8.7.28
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.
28. ∫ ln² x dx
Problem 8.2.1
1. On which derivative rule is integration by parts based?
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 8.6.19
7–84. Evaluate the following integrals.
19. ∫ from 0 to π/2 [sin⁷x] 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
Problem 8.5.73
73. Two methods Evaluate ∫ dx/(x² - 1), for x > 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.
Problem 8.3.47
9–61. Trigonometric integrals Evaluate the following integrals.
47. ∫ (csc⁴x)/(cot²x) dx
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 8.2.79
79. Tabular integration extended Refer to Exercise 77.
a. The following table shows the method of tabular integration applied to
∫ eˣ cos x dx.
Use the table to express ∫ eˣ cos x dx in terms of the sum of functions and an indefinite integral.
b. Solve the equation in part (a) for ∫ eʳ cos z dz.
c. Evaluate ∫ e⁻ᶻ sin 3z dz by applying the idea from parts (a) and (b).
Problem 8.9.104
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.
104. f(t) = t → F(s) = 1/s²
Problem 8.1.53
7–64. Integration review Evaluate the following integrals.
53. ∫ eˣ sec(eˣ + 1) dx
Problem 8.4.83
Visual proof Let F(x)=∫₀ˣ √(a²−t²) dt. The figure shows that F(x)= area of sector OAB+ area of triangle OBC.
a. Use the figure to prove that
F(x) = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2
b. Conclude that ∫ √(a²−x²) dx = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2 + C.
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 8.7.11
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.
11. ∫ 3u / (2u + 7) du
Problem 8.8.64
64. (Use of Tech) Normal distribution of movie lengths
A study revealed that the lengths of U.S. movies are normally distributed with a mean of 110 minutes and a standard deviation of 22 minutes. This means that the fraction of movies with lengths between a and b minutes (with a < b) is given by the integral:
(1/(22√(2π))) ∫[a to b] e^(-((x-110)/22)²/2) dx.
What percentage of U.S. movies are between 1 hr and 1.5 hr long (60-90 min)?
Problem 8.5.54
23-64. Integration Evaluate the following integrals.
54. ∫ (z + 1)/[z(z² + 4)] dz
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 8.1.10
7–64. Integration review Evaluate the following integrals.
10. ∫ e^(3 - 4x) dx
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
