12. Techniques of Integration

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?

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

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

14. ∫ s · e⁻²ˢ ds

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