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Problem 8.4.51
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
51. ∫ x²/√(4 + x²) dx
Problem 8.1.4
Let f(x) = (4x³ + x² + 4x + 2) / (x² + 1). Use long division to show that f(x) = 4x + 1 + 1 / (x² + 1) and use this result to evaluate ∫f(x) dx.
Problem 8.3.26
9–61. Trigonometric integrals Evaluate the following integrals.
26. ∫ sin³x cos³ᐟ²x dx
Problem 8.7.58
49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.
58. ∫₀^{2π} dt / (4 + 2 sin t)²
Problem 8.7.52
49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.
52. ∫ from 0 to π/2 of cos⁶x dx
Problem 8.2.26
9–40. Integration by parts Evaluate the following integrals using integration by parts.
26. ∫ t³ sin(t) dt
Problem 8.7.74
71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
74. ∫xⁿ arcsin(x) dx (Hint: integration by parts.)
Problem 8.4.36
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
36. ∫[8√2 to 16] 1/√(x² - 64) dx
Problem 8.2.69
69. Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, π]. Which is greater, the volume when R is revolved about the x-axis, or the volume when R is revolved about the y-axis?
Problem 8.8.60
58–61. {Use of Tech} Using Simpson's Rule Approximate the following integrals using Simpson's Rule. Experiment with values of n to ensure the error is less than 10⁻³.
60. ∫(from 0 to π) ln(2 + cos x) dx = π ln((2 + √3)/2)
Problem 8.5.29
23-64. Integration Evaluate the following integrals.
29. ∫₋₁² [(5x) / (x² - x - 6)] dx
Problem 8.7.42
41–48. Geometry problems Use a table of integrals to solve the following problems.
42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.
Problem 8.5.50
23-64. Integration Evaluate the following integrals.
50. ∫ 8(x² + 4)/[x(x² + 8)] dx
Problem 8.3.53
9–61. Trigonometric integrals Evaluate the following integrals.
53. ∫ from 0 to π/4 of sec⁴θ dθ
Problem 8.6.41
7–84. Evaluate the following integrals.
41. ∫ cot^(3/2)x · csc⁴x dx
Problem 8.2.4
4. How is integration by parts used to evaluate a definite integral?
Problem 8.9.36
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
36. ∫ (from e² to ∞) 1/(x lnᵖ x) dx, p > 1
Problem 8.4.26
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x dx
Problem 8.6.13
7–84. Evaluate the following integrals.
13. ∫ [1 / (eˣ √(1 – e²ˣ))] dx
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 8.4.46
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
46. ∫ 1/√(1 - 2x²) dx
Problem 8.3.45
9–61. Trigonometric integrals Evaluate the following integrals.
45. ∫ sec²x tan¹ᐟ²x dx
Problem 8.6.74
Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx
Problem 8.1.47
7–64. Integration review Evaluate the following integrals.
47. ∫ dx / (x⁻¹ + 1)
Problem 8.R.101
101. Comparing volumes Let R be the region bounded by the graph of y = sin(x) and the x-axis on the interval [0, π]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or about the y-axis?
Problem 8.R.57
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
57. ∫ (from 0 to √3/2) 4/(9 + 4x²) dx
Problem 8.R.46
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
46. ∫ (x³ + 4x² + 12x + 4)/((x² + 4x + 10)²) dx
Problem 8.R.92
92. Integral with a parameter For what values of p does the integral
∫ (from 1 to ∞) dx/xlnᵖ(x) converge, and what is its value (in terms of p)?
Problem 8.R.123
123. Region between curves Find the area of the region bounded by the graphs of y = tan(x) and y = sec(x) on the interval [0, π/4].
Problem 8.R.54
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
54. ∫ dx/√(9x² - 25), x > 5/3