7. Antiderivatives & Indefinite Integrals

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)

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

Variations on the substitution method Evaluate the following integrals.                                                                                                        

 ∫ (eˣ ― e⁻ˣ)/ (eˣ + e⁻ˣ) d𝓍

Variations on the substitution method Evaluate the following integrals.                                                                                                        

 ∫ (𝒵 + 1) √(3𝒵 + 2) d𝒵

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

  ∫ 𝓍 sin⁴ 𝓍² cos 𝓍² d𝓍 (Hint: Begin with u = 𝓍², and then use v = sin u .)

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

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

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.

 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.