12. Techniques of Integration

7–84. Evaluate the following integrals.

22. ∫ [1 / ((x - a)(x - b))] dx, where a ≠ b

7–84. Evaluate the following integrals.

59. ∫ 1/(x⁴ + x²) dx

7–84. Evaluate the following integrals.

47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

15. ∫ (from 1 to 2) (3x⁵ + 48x³ + 3x² + 16)/(x³ + 16x) dx

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

71. ∫ (2x² - 4x)/(x² - 4) dx

63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.

∫₋₂² dt/(t² – 9)

17-22. Give the partial fraction decomposition for the following expressions.

17. (5x - 7) / (x² - 3x + 2)

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

15. x / ((x⁴ - 16)²)

23-64. Integration Evaluate the following integrals.

23. ∫ [3 / ((x - 1)(x + 2))] dx

23-64. Integration Evaluate the following integrals.

26. ∫₀¹ [1 / (t² - 9)] dt

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

6. (4x + 1)/(4x² - 1)

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

9. 4/(x⁵ - 5x³ + 4x)

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))

23-64. Integration Evaluate the following integrals.

29. ∫₋₁² [(5x) / (x² - x - 6)] dx

23-64. Integration Evaluate the following integrals.

32. ∫ (4x - 2)/(x³ - x) dx