8. Definite Integrals

7–84. Evaluate the following integrals.

7. ∫ from 0 to π/2 [sin θ / (1 + cos² θ)] dθ

9–61. Trigonometric integrals Evaluate the following integrals.

22. ∫[π/4 to π/2] sin²(2x) cos³(2x) dx

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

9. ∫ (from 0 to π/4) cos⁵ 2x sin² 2x 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)

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

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

 ∫₁/₃^¹/√³ 4/(9𝓍² + 1) d𝓍

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

 ∫₀ᵉ² (ln p)/p dp

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

 ∫₋₁¹ (𝓍―1) (𝓍²―2𝓍)⁷ d𝓍

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

 ∫₀^π/⁴ eˢᶦⁿ² ˣ sin 2𝓍 d𝓍

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

26. ∫[√2 to √2] √(x² - 1)/x dx

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

36. ∫[8√2 to 16] 1/√(x² - 64) dx

Evaluating integrals Evaluate the following integrals.

∫₁ᵉ d𝓍 / [𝓍(1 + ln 𝓍)]

Evaluating integrals Evaluate the following integrals.

∫₀¹ 𝓍 • 2ˣ²⁺¹ d𝓍

Evaluating integrals Evaluate the following integrals.

∫₀² (2𝓍 + 1)³ d𝓍

Evaluating integrals Evaluate the following integrals.

∫₀^²π cos² 𝓍/6 d𝓍