8. Definite Integrals

9–61. Trigonometric integrals Evaluate the following integrals.

59. ∫ from 0 to π/2 of √(1 - cos2x) dx

7–84. Evaluate the following integrals.

56. ∫ from π to 3π/2 sin2x e^(sin²x) dx

Evaluate the integrals in Exercises 41–60.

57. ∫(from 1 to 2)cosh(ln t)/t dt

Evaluate the integrals in Exercises 47–68.

∫₀¹ dr .

∛(7 - 5r)²

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

∫₀³ (2x - 1) / (x + 1) dx

Evaluate the integrals in Exercises 47–68.

∫₀ ^π/2 5(sin x)³/² cos x dx

7–84. Evaluate the following integrals.

62. ∫ from 0 to π/2 √(1 + cosθ) dθ

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

∫₋₂² (e^{z/2}) / (e^{z/2} + 1) dz

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

∫₀³ (x + 2)√(x + 1) dx

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫(from π/2 to 2π/3) cos θ dθ / (sin θ cos θ + sin θ)

Evaluate the indefinite integral.

${\int }_{}^{}x{(5+x)}^{79}dx$

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)

Evaluate the integrals in Exercises 1–22.

∫₀^(π/2) sin²(2θ) cos³(2θ) dθ

Evaluate the integrals in Exercises 1–22.

∫₀^π 3sin(x/3) dx

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

26. ∫[√2 to √2] √(x² - 1)/x dx