12. Techniques of Integration

Evaluate the indefinite integral.

${\int }_{}^{}{x}^2\cos 4xdx$

Evaluate the definite integral.

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

Evaluate the definite integral.

${\int }_0^{\pi }{t}^2\sin tdt$

92–98. Evaluate the following integrals.

92. ∫[1 to √2] y⁸ e^(y²) dy

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

28. ∫ ln² x dx

71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.

74. ∫xⁿ arcsin(x) dx (Hint: integration by parts.)

7–84. Evaluate the following integrals.

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

7–84. Evaluate the following integrals.

79. ∫ (arcsinx)/x² dx

92–98. Evaluate the following integrals.

97. ∫ tan⁻¹(∛x) dx

7–84. Evaluate the following integrals.

77. ∫ arccosx dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

51. ∫ x²/√(4 + x²) dx

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

11. ∫ t · e⁶ᵗ dt

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

17. ∫ x · 3x dx

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

20. ∫ sin⁻¹(x) dx

1. On which derivative rule is integration by parts based?