7. Antiderivatives & Indefinite Integrals

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 .)

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (sec² Θ + sec Θ tan Θ)dΘ

Use any method to evaluate the integrals in Exercises 55–66.

∫ x / (x + √(x² + 2)) dx

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ (𝓍³ + 5𝓍 ―7) d𝓍

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ y² /(y³ + 27) dy

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.

42. ∫ 1/(x²√(9x² - 1)) dx, x > 1/3

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

117. ∫ dr / (1 + √r)

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.

7–84. Evaluate the following integrals.

25. ∫ [1 / (x√(1 - x²))] dx

Evaluate the integrals in Exercises 37–46.

∫(sin 2θ - cos 2θ)/(sin 2θ + cos 2θ)³dθ

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.)

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ x^2 √(2x - x^2) dx

59. Two Methods

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

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (√x(2x⁶ - 4³√)dx