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Problem 8.5.41
23-64. Integration Evaluate the following integrals.
41. ∫₋₁¹ x/(x + 3)² dx
Problem 8.5.60
23-64. Integration Evaluate the following integrals.
60.∫ 1/[(y² + 1)(y² + 2)] dy
Problem 8.5.62
23-64. Integration Evaluate the following integrals.
62. ∫ 1/[(x + 1)(x² + 2x + 2)²] dx
Problem 8.5.88
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).
Problem 8.5.91
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).
Problem 8.5.85
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
Problem 8.5.82
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))]
Problem 8.5.79
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
Problem 8.5.76
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
Problem 8.5.65a
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.
Problem 8.5.66
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
Problem 8.5.70
69-72. Volumes of solids Find the volume of the following solids.
70. The region bounded by y = 1/[x²(x² + 2)²], y = 0, x = 1, and x = 2 is revolved about the y-axis.
Problem 8.5.73
73. Two methods Evaluate ∫ dx/(x² - 1), for x > 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.
Problem 8.5.93b
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?
Problem 8.5.93d
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?
Problem 8.5.2b
2. Give an example of each of the following.
b. A repeated linear factor
Problem 8.5.2d
2. Give an example of each of the following.
d. A repeated irreducible quadratic factor
Problem 8.5.3c
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
Problem 8.5.20
17-22. Give the partial fraction decomposition for the following expressions.
20. (x² - 4x + 11) / ((x - 3)(x - 1)(x + 1))
Problem 8.5.38
23-64. Integration Evaluate the following integrals.
38. ∫₀⁵ 2/(x² - 4x - 32) dx
Problem 8.5.44
23-64. Integration Evaluate the following integrals.
44. ∫₁² 2/[t³(t + 1)] dt
Problem 8.5.47
23-64. Integration Evaluate the following integrals.
47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx
Problem 8.5.50
23-64. Integration Evaluate the following integrals.
50. ∫ 8(x² + 4)/[x(x² + 8)] dx
Problem 8.5.54
23-64. Integration Evaluate the following integrals.
54. ∫ (z + 1)/[z(z² + 4)] dz
Problem 8.5.57
23-64. Integration Evaluate the following integrals.
57. ∫ (x³ + 5x)/(x² + 3)² dx
Problem 8.5.96
96. Challenge
Show that with the change of variables u = √tan x, the integral
∫ √tan x dx
can be converted to an integral amenable to partial fractions. Evaluate
∫[0 to π/4] √tan x dx.
Problem 8.5.94c
94. [Use of Tech] Skydiving A skydiver has a downward velocity given by v(t) = V_T [(1 - e^(-2gt/V_T))/(1 + e^(-2gt/V_T))],
where t = 0 is the instant the skydiver starts falling, g = 9.8 m/s² is the acceleration due to gravity, and V_T is the terminal velocity of the skydiver.
c. Verify by integration that the position function is given by
s(t) = V_T t + (V_T²/g) ln[(1 + e^(-2gt/V_T))/2],
where s'(t) = v(t) and s(0) = 0.
Problem 8.6.45
7–84. Evaluate the following integrals.
45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx
Problem 8.6.11
7–84. Evaluate the following integrals.
11. ∫ from 0 to π/4 (sec x – cos x)² dx
Problem 8.6.7
7–84. Evaluate the following integrals.
7. ∫ from 0 to π/2 [sin θ / (1 + cos² θ)] dθ