12. Techniques of Integration

In Exercises 67–73, use integration by parts to establish the reduction formula.

∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx

Finding volume

Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.

Find the volume of the solid generated by revolving R about

a. the x-axis.

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

Evaluate the integrals in Exercises 1–6.

∫ (arcsin x)² dx

Evaluate the integrals in Exercises 1–8 using integration by parts.

∫ x² sin(1 − x) dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ (x² - 2x + 1) e^(2x) dx

2. When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ∫ f(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

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ x·e^(2x) 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.

79. ∫ x sin⁻¹(2x) dx

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ x² tan⁻¹(x / 2) dx

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

40. ∫ e^√x dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ p⁴ e^(-p) dp

Evaluate the integrals in Exercises 1–8 using integration by parts.

∫ x sin(x) cos(x) dx

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