8. Definite Integrals

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ from 0 to 1 x√(1 - x) dx

Evaluate the integrals in Exercises 1–22.

∫₀^π sin⁵(x/2) dx

Evaluate the indefinite integral.

${\int }_{}^{}\frac{t}{\sqrt{t-2}}dt$

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫ₑᵉ^³ dx / (x ln x ln²(ln x))

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

  ∫₀^π/² (cos θ sin θ) / √(cos² θ + 16) dθ (Hint: Begin with u = cos θ .)

7–64. Integration review Evaluate the following integrals.

32. ∫ from 0 to 2 of x / (x² + 4x + 8) dx

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

68. ∫ (from -1 to 1) dx/(x² + 2x + 5)

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫₋₁³ (4x² - 7) / (2x + 3) dx

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

9. ∫[5 to 5√3] √(100 - x²) dx

Evaluate the integrals in Exercises 39–56.

45. ∫(from 1 to 2)(2ln x)/x dx

Evaluate the integrals in Exercises 1–22.

∫₀^π 8cos⁴(2πx) dx

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

51. ∫ (from 0 to π/4) sin⁵(4θ) dθ

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ dy / (y√(1 + (ln y)²)) from 1 to e

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

 ∫₀ᵉ² (ln p)/p dp

Change of variables Use the change of variables u³ = 𝓍² ― 1 to evaluate the integral ∫₁³ 𝓍∛(𝓍²―1) d𝓍 .