12. Techniques of Integration

9–40. Integration by parts Evaluate the following integrals using integration by parts.

32. ∫ from 0 to 1 x² 2ˣ dx

9–40. Integration by parts Evaluate the following integrals using integration by parts.

36. ∫ from 0 to ln2 x eˣ dx

9–40. Integration by parts Evaluate the following integrals using integration by parts.

38. ∫ x² ln²(x) dx

9–40. Integration by parts Evaluate the following integrals using integration by parts.

40. ∫ e^√x dx

48. Integral of sec³x Use integration by parts to show that:

∫ sec³x dx = (1/2) secx tanx + (1/2) ∫ secx dx

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

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

54-57. Applying Reduction Formulas Use the reduction formulas from Exercises 50-53 to evaluate the following integrals:

55. ∫ x² cos(5x) dx

58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:

b. Use substitution.

60. Two Methods

a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.

60. Two Methods

c. Verify that your answers to parts (a) and (b) are consistent.

62. Two integration methods Evaluate ∫ sin x cos x dx using integration by parts. Then evaluate the integral using a substitution. Reconcile your answers

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

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

43. ∫ eˣ sin(x) dx

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.

82. ∫ (sin⁻¹(ax)) / x² dx, a > 0