7. Antiderivatives & Indefinite Integrals

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.

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ sin 𝓍 sec⁸ 𝓍 d𝓍

37–56. Integrals Evaluate each integral.

∫ dx/x√(16 + x²)

59. Two Methods

b. Evaluate ∫(x / √(x + 1)) dx using substitution.

General results Evaluate the following integrals in which the function ƒ is unspecified. Note that ƒ⁽ᵖ⁾ is the pth derivative of ƒ and ƒᵖ is the pth power of ƒ. Assume ƒ and its derivatives are continuous for all real numbers. 

∫ (5 ƒ³ (𝓍) + 7ƒ² (𝓍) + ƒ (𝓍 )) ƒ'(𝓍) d𝓍

81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

a. I₀ = ∫ e⁻ˣ² dx cannot be expressed in terms of elementary functions. Evaluate I₁.

74. A secant reduction formula

Prove that for positive integers n ≠ 1,

∫ secⁿ x dx = (secⁿ⁻² x tan x)/(n − 1) + (n − 2)/(n − 1) ∫ secⁿ⁻² x dx.

(Hint: Integrate by parts with u = secⁿ⁻² x and dv = sec² x dx.)