12. Techniques of Integration

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

65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. To evaluate ∫ (4x⁶)/(x⁴ + 3x²) dx, the first step is to find the partial fraction decomposition of the integrand.

73. Two methods Evaluate ∫ dx/(x² - 1), for x > 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

66-68. Areas of regions (Use of Tech) Find the area of the following regions.

66. The region bounded by the curve y = (x - x²)/[(x + 1)(x² + 1)] and the x-axis from x = 0 to x = 1

93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:

vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).

b. Which car travels farthest on the interval 0 ≤ t ≤ 5?

93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:

vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).

d. Which car ultimately gains the lead and remains in front?

2. Give an example of each of the following.

b. A repeated linear factor

2. Give an example of each of the following.

d. A repeated irreducible quadratic factor

3. What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following?

c. A factor of (x² + 2x + 6) in the denominator

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

20. (x² - 4x + 11) / ((x - 3)(x - 1)(x + 1))

23-64. Integration Evaluate the following integrals.

38. ∫₀⁵ 2/(x² - 4x - 32) dx

23-64. Integration Evaluate the following integrals.

44. ∫₁² 2/[t³(t + 1)] dt

23-64. Integration Evaluate the following integrals.

47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx

23-64. Integration Evaluate the following integrals.

50. ∫ 8(x² + 4)/[x(x² + 8)] dx