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Problem 8.8.6
5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.
6. T(4)
Problem 8.2.23
9–40. Integration by parts Evaluate the following integrals using integration by parts.
23. ∫ x² sin(2x) dx
Problem 8.3.16
9–61. Trigonometric integrals Evaluate the following integrals.
16. ∫ sin²θ cos⁵θ dθ
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 8.1.57
7–64. Integration review Evaluate the following integrals.
57. ∫ dx / (x¹⸍² + x³⸍²)
Problem 8.1.53
7–64. Integration review Evaluate the following integrals.
53. ∫ eˣ sec(eˣ + 1) dx
Problem 8.4.62
60–69. Completing the square Evaluate the following integrals.
62. ∫ du / (2u² - 12u + 36)
Problem 8.5.9
5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
9. 4/(x⁵ - 5x³ + 4x)
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 8.9.13
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
13. ∫ (from 0 to ∞) cos x 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 8.5.23
23-64. Integration Evaluate the following integrals.
23. ∫ [3 / ((x - 1)(x + 2))] dx
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 8.6.30
7–84. Evaluate the following integrals.
30. ∫ from 5/2 to 5√3/2 [1 / (v² √(25 - v²))] dv
Problem 8.4.51
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
51. ∫ x²/√(4 + x²) 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 8.9.98
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 8.6.49
7–84. Evaluate the following integrals.
49. ∫ tan³x · sec⁹x dx
Problem 8.4.33
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
33. ∫ √(x² - 9)/x dx, x > 3
Problem 8.4.19
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
19. ∫ 1/√(x² - 81) dx, x > 9
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 8.6.27
7–84. Evaluate the following integrals.
27. ∫ sin⁴(x/2) dx
Problem 8.6.19
7–84. Evaluate the following integrals.
19. ∫ from 0 to π/2 [sin⁷x] dx
Problem 8.6.16
7–84. Evaluate the following integrals.
16. ∫ [1 / (x⁴ – 1)] dx
Problem 8.5.96
96. Challenge
Show that with the change of variables u = √tan x, the integral
∫ √tan x dx
can be converted to an integral amenable to partial fractions. Evaluate
∫[0 to π/4] √tan x dx.
Problem 8.5.15
5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
15. x / ((x⁴ - 16)²)
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 8.3.45
9–61. Trigonometric integrals Evaluate the following integrals.
45. ∫ sec²x tan¹ᐟ²x dx
