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

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

49. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample:

c. ∫ v du = u·v - ∫ u dv

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

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.