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Problem 8.1.68
68. Different methods
a. Evaluate ∫(cot x csc² x) dx using the substitution u=cotx.
Problem 8.1.1
What change of variables would you use for the integral ∫(4 - 7x)^(-6) 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 8.1.16
7–64. Integration review Evaluate the following integrals.
16. ∫ from 0 to 1 of (t² / (1 + t⁶)) dt
Problem 8.1.18
7–64. Integration review Evaluate the following integrals.
18. ∫ from 3 to 7 of (t - 6) * √(t - 3) dt
Problem 8.1.20
7–64. Integration review Evaluate the following integrals.
20. ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) dx
Problem 8.1.24
7–64. Integration review Evaluate the following integrals.
24. ∫ from 0 to θ of (x⁵⸍² - x¹⸍²) / x³⸍² dx
Problem 8.1.28
7–64. Integration review Evaluate the following integrals.
28. ∫ (3x + 1) / √(4 - x²) dx
Problem 8.1.32
7–64. Integration review Evaluate the following integrals.
32. ∫ from 0 to 2 of x / (x² + 4x + 8) dx
Problem 8.1.36
7–64. Integration review Evaluate the following integrals.
36. ∫ (t³ - 2) / (t + 1) dt
Problem 8.1.38
7–64. Integration review Evaluate the following integrals.
38. ∫ x / (x⁴ + 2x² + 1) dx
Problem 8.1.40
7–64. Integration review Evaluate the following integrals.
40. ∫ (1 - x) / (1 - √x) dx
Problem 8.1.44
7–64. Integration review Evaluate the following integrals.
44. ∫ from 0 to √3 of (6x³) / √(x² + 1) dx
Problem 8.1.47
7–64. Integration review Evaluate the following integrals.
47. ∫ dx / (x⁻¹ + 1)
Problem 8.1.49
7–64. Integration review Evaluate the following integrals.
49. ∫ √(9 + √(t + 1)) dt
Problem 8.1.51
7–64. Integration review Evaluate the following integrals.
51. ∫ from -1 to 0 of x / (x² + 2x + 2) dx
Problem 8.1.53
7–64. Integration review Evaluate the following integrals.
53. ∫ eˣ sec(eˣ + 1) dx
Problem 8.1.57
7–64. Integration review Evaluate the following integrals.
57. ∫ dx / (x¹⸍² + x³⸍²)
Problem 8.1.60
7–64. Integration review Evaluate the following integrals.
60. ∫ from 0 to π/4 of 3√(1 + sin 2x) dx
Problem 8.1.69
69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.
Problem 8.1.70
70. Different methods Let I=∫(x+2)/(x+4)dx.
b. Evaluate I without performing long division on the integrand.
Problem 8.1.76b
76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x
Problem 8.1.62
7–64. Integration review Evaluate the following integrals.
62. ∫ (-x⁵ - x⁴ - 2x³ + 4x + 3) / (x² + x + 1) dx
Problem 8.1.68b
68. Different methods
b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.
Problem 8.1.74
74. Volume of a Solid
Consider the region R bounded by:
The graph of f(x) = 1/(x + 2)
The x-axis on the interval [0,3].
Find the volume of the solid formed when R is revolved about the y-axis.
Problem 8.1.77
Let f(x) = √(x + 1). Find the area of the surface generated when:
Region bounded by f(x) and the x-axis on [0, 1]
Revolved about the x-axis
Problem 8.1.65a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. ∫(3/(x² + 4)) dx = ∫(3/x²) dx + ∫(3/4) dx.
Problem 8.1.65d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. ∫(1/eˣ) dx = ln eˣ + C.
Problem 8.2.49c
49. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample:
c. ∫ v du = u·v - ∫ u dv
Problem 8.2.44
42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.