Back65. 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
23-64. Integration Evaluate the following integrals.
54. ∫ (z + 1)/[z(z² + 4)] dz
23-64. Integration Evaluate the following integrals.
57. ∫ (x³ + 5x)/(x² + 3)² dx