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

9–61. Trigonometric integrals Evaluate the following integrals.

60. ∫ from 0 to π/8 of √(1 - cos8x) dx

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

∫₀^(π/3) tan³x·sec²x dx

Evaluate the following integrals.

65. ∫ from 0 to 1/6 1/√(1 - 9x²) dx

37–56. Integrals Evaluate each integral.

∫₂₅²²⁵ dx / (x² + 25x) (Hint: √(x² + 25x) = √x √(x + 25).)

Evaluate the integrals in Exercises 47–68.

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∫₁⁴ (1 + √u)¹/² du

√u

The composite function ƒ(g(𝓍)) consists of an inner function g and an outer function ƒ. If an integrand includes ƒ(g(𝓍)), which function is often a likely choice for a new variable u?

2. What change of variables is suggested by an integral containing √(x² + 36)?

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.

∫ (1 / (cos² x tan x)) dx from π/3 to π/4

Evaluate the definite integral.

${\int }_0^1\frac{t}{\sqrt{t^2+1}}dt$

7–84. Evaluate the following integrals.

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

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

36. ∫[8√2 to 16] 1/√(x² - 64) dx

135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:

(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.

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

57. ∫ (from 0 to √3/2) 4/(9 + 4x²) dx

When using a change of variables u = g(𝓍) to evaluate the definite integral ∫ₐᵇ ƒ(g(𝓍)) g' (𝓍) d(𝓍), how are the limits of integration transformed?