7. Antiderivatives & Indefinite Integrals

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

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

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

74. ∫ dx/√(√(1 + √x))

76-81. Table of integrals Use a table of integrals to evaluate the following integrals.

76. ∫ x(2x + 3)⁵ dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

30. ∫ x³√(1 - x²) dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

33. ∫ √(x² - 9)/x dx, x > 3

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

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

39. ∫ x²/(100 - x²)^(3/2) dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

42. ∫ 1/(x²√(9x² - 1)) dx, x > 1/3

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

46. ∫ 1/√(1 - 2x²) dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

48. ∫ √(9 - 4x²) dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

54. ∫ y⁴/(1 + y²) dy

57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The integral ∫ dx/(x² + 4x + 9) cannot be evaluated using a trigonometric substitution.

7–84. Evaluate the following integrals.

64. ∫ (ln(ax))/x dx, where a ≠ 0

60–69. Completing the square Evaluate the following integrals.

62. ∫ du / (2u² - 12u + 36)