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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 8.4.74
{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 8.1.77
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 8.7.34
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.
34. ∫ dx / (x(x¹⁰ + 1))
Problem 8.5.54
23-64. Integration Evaluate the following integrals.
54. ∫ (z + 1)/[z(z² + 4)] dz
Problem 8.2.17
9–40. Integration by parts Evaluate the following integrals using integration by parts.
17. ∫ x · 3x dx
Problem 8.5.32
23-64. Integration Evaluate the following integrals.
32. ∫ (4x - 2)/(x³ - x) dx
Problem 8.3.51
9–61. Trigonometric integrals Evaluate the following integrals.
51. ∫ (csc²x + csc⁴x) dx
Problem 8.7.79
79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
79. ∫ x sin⁻¹(2x) dx
Problem 8.4.62
60–69. Completing the square Evaluate the following integrals.
62. ∫ du / (2u² - 12u + 36)
Problem 8.8.30
29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
30. ∫(0 to 6) (x³/16 - x) dx = 4
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 8.1.32
7–64. Integration review Evaluate the following integrals.
32. ∫ from 0 to 2 of x / (x² + 4x + 8) dx
Problem 8.6.77
7–84. Evaluate the following integrals.
77. ∫ arccosx dx
Problem 8.1.4
Let f(x) = (4x³ + x² + 4x + 2) / (x² + 1). Use long division to show that f(x) = 4x + 1 + 1 / (x² + 1) and use this result to evaluate ∫f(x) dx.
Problem 8.3.4
4. Describe the method used to integrate sinᵐx cosⁿx, for m even and n odd.
Problem 8.2.40
9–40. Integration by parts Evaluate the following integrals using integration by parts.
40. ∫ e^√x dx
Problem 8.9.4
4. Evaluate ∫ (from 0 to 1) (1/x^(1/5)) dx after writing the integral as a limit.
Problem 8.3.19
9–61. Trigonometric integrals Evaluate the following integrals.
19. ∫[0 to π/3] sin⁵x cos⁻²x dx
Problem 8.7.58
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.
58. ∫₀^{2π} dt / (4 + 2 sin t)²
Problem 8.2.62
62. Two integration methods Evaluate ∫ sin x cos x dx using integration by parts. Then evaluate the integral using a substitution. Reconcile your answers
Problem 8.5.57
23-64. Integration Evaluate the following integrals.
57. ∫ (x³ + 5x)/(x² + 3)² dx
Problem 8.9.94
94. The family f(x) = 1/xᵖ revisited Consider the family of functions f(x) = 1/xᵖ, where p is a real number.
For what values of p does the integral ∫(1 to ∞) 1/xᵖ dx exist?
What is its value when it exists?
Problem 8.6.62
7–84. Evaluate the following integrals.
62. ∫ from 0 to π/2 √(1 + cosθ) dθ
Problem 8.5.20
17-22. Give the partial fraction decomposition for the following expressions.
20. (x² - 4x + 11) / ((x - 3)(x - 1)(x + 1))
Problem 8.4.12
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
12. ∫[1/2 to 1] √(1 - x²)/x² dx
Problem 8.3.26
9–61. Trigonometric integrals Evaluate the following integrals.
26. ∫ sin³x cos³ᐟ²x dx
Problem 8.1.40
7–64. Integration review Evaluate the following integrals.
40. ∫ (1 - x) / (1 - √x) 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
