12. Techniques of Integration

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

26. ∫ t³ sin(t) dt

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

23. ∫ x² sin(2x) dx

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

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

29. ∫ e⁻ˣ sin(4x) dx

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.