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Problem 8.2.11
9–40. Integration by parts Evaluate the following integrals using integration by parts.
11. ∫ t · e⁶ᵗ dt
Problem 8.2.17
9–40. Integration by parts Evaluate the following integrals using integration by parts.
17. ∫ x · 3x dx
Problem 8.2.20
9–40. Integration by parts Evaluate the following integrals using integration by parts.
20. ∫ sin⁻¹(x) dx
Problem 8.2.1
1. On which derivative rule is integration by parts based?
Problem 8.2.4
4. How is integration by parts used to evaluate a definite integral?
Problem 8.2.14
9–40. Integration by parts Evaluate the following integrals using integration by parts.
14. ∫ s · e⁻²ˢ ds
Problem 8.2.26
9–40. Integration by parts Evaluate the following integrals using integration by parts.
26. ∫ t³ sin(t) dt
Problem 8.2.23
9–40. Integration by parts Evaluate the following integrals using integration by parts.
23. ∫ x² sin(2x) dx
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 8.2.29
9–40. Integration by parts Evaluate the following integrals using integration by parts.
29. ∫ e⁻ˣ sin(4x) dx
Problem 8.2.32
9–40. Integration by parts Evaluate the following integrals using integration by parts.
32. ∫ from 0 to 1 x² 2ˣ dx
Problem 8.2.36
9–40. Integration by parts Evaluate the following integrals using integration by parts.
36. ∫ from 0 to ln2 x eˣ dx
Problem 8.2.38
9–40. Integration by parts Evaluate the following integrals using integration by parts.
38. ∫ x² ln²(x) dx
Problem 8.2.40
9–40. Integration by parts Evaluate the following integrals using integration by parts.
40. ∫ e^√x dx
Problem 8.2.42
42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
42. The region bounded by f(x) = ln(x), y = 1, and the coordinate axes is revolved about the x-axis.
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 8.2.51
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 8.2.53
50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:
53. ∫ lnⁿ(x) dx = x lnⁿ(x) - n ∫ lnⁿ⁻¹(x) dx
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 8.2.58b
58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:
b. Use substitution.
Problem 8.2.59b
59. Two Methods
b. Evaluate ∫(x / √(x + 1)) dx using substitution.
Problem 8.2.60a
60. Two Methods
a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.
Problem 8.2.60c
60. Two Methods
c. Verify that your answers to parts (a) and (b) are consistent.
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 8.2.66
66. Integrating derivatives
Use integration by parts to show that if f' is continuous on [a, b], then
∫[a to b] f(x)f'(x) dx = (1/2)[f(b)² - f(a)²]
Problem 8.2.69
69. Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, π]. Which is greater, the volume when R is revolved about the x-axis, or the volume when R is revolved about the y-axis?
Problem 8.2.72
72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].
Problem 8.2.78e
78. Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).
e. ∫ (2x² - 3x) / (x - 1)³ 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 8.2.77d
77. Tabular integration Consider the integral ∫ f(x)g(x) dx, where f can be differentiated repeatedly and g can be integrated repeatedly
Let Gₖ represent the result of calculating k indefinite integrals of g (omitting constants of integration).
d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts.
Evaluate ∫ x² e^(x/2) dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in.
