7. Antiderivatives & Indefinite Integrals

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

7–84. Evaluate the following integrals.

64. ∫ (ln(ax))/x dx, where a ≠ 0

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

44. ∫ 1/√(16 + 4x²) dx

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

68. ∫ dx / sqrt((x - 1)(3 - x))

Variations on the substitution method Evaluate the following integrals.                                                                                                        

 ∫ 𝓍/(∛𝓍 + 4) d𝓍

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

  ∫ d𝓍 / [√1 + √(1 + 𝓍)] (Hint: Begin with u = √(1 + 𝓍 .)