12. Techniques of Integration

4. How is integration by parts used to evaluate a definite integral?

60. Two Methods

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

Finding volume: Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve y = x sin(x), 0 ≤ x ≤ π, about

a. The y-axis.

(See Exercise 57 for a graph.)

7–84. Evaluate the following integrals.

79. ∫ (arcsinx)/x² 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

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

∫ (r² + r + 1) e^r dr

7–84. Evaluate the following integrals.

38. ∫ from π/6 to π/2 [cos x · ln(sin x)] dx

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

∫ √x e√x dx

Equations (4) and (5) lead to different formulas for the integral of arctan x:

a. ∫ arctan x dx = x arctan x - ln sec(arctan x) + C [Eq. (4)]

b. ∫ arctan x dx = x arctan x - ln √(1 + x²) + C [Eq. (5)]

Can both integrations be correct? Explain.

Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)

To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.

∫ log₂ x dx

Evaluate the definite integral.

${\int }_1^{2}x\ln xdx$

92–98. Evaluate the following integrals.

97. ∫ tan⁻¹(∛x) dx

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

∫ x⁵ e³ˣ dx

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ ln(x + x²) dx

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

∫ e^(-y) cos(y) dy