8. Definite Integrals

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

∫₀³ (2x - 1) / (x + 1) dx

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

∫₋₂² (e^{z/2}) / (e^{z/2} + 1) dz

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

∫₀^{π} 2^{sin x} · cos x dx

37–56. Integrals Evaluate each integral.

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

66. Integrating derivatives

Use integration by parts to show that if f' is continuous on [a, b], then

∫[a to b] f(x)f'(x) dx = (1/2)[f(b)² - f(a)²]

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

∫ₑᵉ^³ dx / (x ln x ln²(ln x))

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

∫₁ᵉ^² (ln x)^5 / x dx

63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.

∫₁/₈¹ dx/x√(1 + x²/³)