8. Definite Integrals

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)

"Electric field due to a line of charge A total charge of Q is distributed uniformly on a line segment of length 2L along the y-axis (see figure). The x-component of the electric field at a point (a, 0) is given by

Eₓ(a) = (kQa/2L) ∫-L L dy/(a² + y²)^(3/2),

where k is a physical constant and a > 0.

a. Confirm that Eₓ(a)=kQ / a √(a²+L²)

b. Letting ρ=Q / 2 L be the charge density on the line segment, show that if L → ∞, then Eₓ(a) = 2kρ / a.

Evaluate the integrals in Exercises 31–78.

39. ∫(from 0 to π)tan(x/3)dx

Evaluate the integrals in Exercises 39–56.

45. ∫(from 1 to 2)(2ln x)/x dx

Evaluate the integrals in Exercises 39–56.

47. ∫(from 2 to 4)dx/(x(ln x)²)

Evaluate the integrals in Exercises 39–56.

52. ∫(from π/4 to π/2)cot(t)dt

Evaluate the integrals in Exercises 1–22.

∫₀^π 8 sin⁴(y) cos²(y) dy

Evaluate the integrals in Exercises 47–68.

_

∫₁⁴ (1 + √u)¹/² du

√u

Evaluate the integrals in Exercises 47–68.

∫₀¹ dr .

∛(7 - 5r)²