8. Definite Integrals

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

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

 ∫ 𝓍 cos²𝓍² d𝓍

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           

(a) ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍 = ½ (ƒ(𝓍))² + C.