12. Techniques of Integration

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

23-64. Integration Evaluate the following integrals.

35. ∫ (x² + 12x - 4)/(x³ - 4x) dx

23-64. Integration Evaluate the following integrals.

41. ∫₋₁¹ x/(x + 3)² dx

23-64. Integration Evaluate the following integrals.

60.∫ 1/[(y² + 1)(y² + 2)] dy

23-64. Integration Evaluate the following integrals.

62. ∫ 1/[(x + 1)(x² + 2x + 2)²] dx

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

88. Evaluate ∫ dx/(2 + cos x).

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

91. Evaluate ∫[0 to π/2] dθ/(cos θ + sin θ).

85. Another form of ∫ sec x dx

a. Verify the identity:

sec x = cos x / (1 - sin² x)

b. Use the identity in part (a) to verify that:

∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

82. ∫ [dx / (x√(1 + 2x))]

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

79. ∫ [sec t / (1 + sin t)] dt

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

76. ∫ [cosθ / (sin³θ - 4sinθ)] dθ

7–84. Evaluate the following integrals.

16. ∫ [1 / (x⁴ – 1)] dx