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Problem 3.3.32
9–61. Trigonometric integrals Evaluate the following integrals.
32. ∫ cot⁵(3x) dx
Problem 5.4.2
If ƒ is an even function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 2 ∫₀ᵃ ƒ(𝓍) d𝓍?
Problem 8.R.60
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
60. ∫ x² coshx dx
Problem 8.RE.6
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
6. ∫ (2 − sin 2θ)/cos² 2θ dθ
Problem 8.RE.22
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
22. ∫ tan³ 5θ dθ
Problem 8.RE.29
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
29. ∫ cos⁴ x/sin⁶ x dx
Problem 8.RE.32
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
32. ∫ csc²(6x) cot(6x) dx
Problem 8.RE.48
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
48. ∫ sin(3x) cos⁶(3x) dx
Problem 8.RE.51
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
51. ∫ (from 0 to π/4) sin⁵(4θ) dθ
Problem 8.R.1b
Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
b. To evaluate the integral ∫dx/√(x² − 100) analytically, it is best to use partial fractions.
Problem 8.R.1e
Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
e. The best approach to evaluating ∫(x³ + 1)/(3x²) dx is to use the change of variables u = x³ + 1.
Problem 8.R.9
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
9. ∫ (from 0 to π/4) cos⁵ 2x sin² 2x dx
Problem 8.R.68
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
68. ∫ (from -1 to 1) dx/(x² + 2x + 5)
Problem 8.R.3
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
3. ∫ (3x)/√(x + 4) dx
Problem 8.R.35
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
35. ∫ x³/√(4x² + 16) dx
Problem 8.R.40
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
40. ∫ (x² - 4)/(x + 4) 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.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
Problem 8.R.74
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
74. ∫ dx/√(√(1 + √x))
Problem 8.R.76
76-81. Table of integrals Use a table of integrals to evaluate the following integrals.
76. ∫ x(2x + 3)⁵ dx
Problem 8.R.79
76-81. Table of integrals Use a table of integrals to evaluate the following integrals.
79. ∫ sec⁵x dx
Problem 8.R.15
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
15. ∫ (from 1 to 2) (3x⁵ + 48x³ + 3x² + 16)/(x³ + 16x) dx
Problem 8.R.18
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
18. ∫ (from 0 to √2) (x + 1)/(3x² + 6) dx
Problem 8.R.25
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
25. ∫ (from -3/2 to -1) dx/(4x² + 12x + 10)
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.1d
Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
d. ∫2 sin x cos x dx = −(1/2) cos 2x + C.
Problem 8.R.10
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
10. ∫ (x³ + 3x² + 1)/(x³ + 1) dx
Problem 8.R.108
108. Arc length Find the length of the curve y = ln(x) on the interval [1, e^2].
Problem 8.R.114a
114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).
a. Find the length of the curve from x = 1 to x = a and call it L(a).
(Hint: The change of variables u = sqrt(x^2 + 1) allows evaluation by partial fractions.)
Problem 8.R.114c
114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).
c. As a increases, L(a) increases as what power of a?