8. Definite Integrals

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

  ∫₀^π/² (cos θ sin θ) / √(cos² θ + 16) dθ (Hint: Begin with u = cos θ .)

Change of variables Use the change of variables u³ = 𝓍² ― 1 to evaluate the integral ∫₁³ 𝓍∛(𝓍²―1) d𝓍 .

On which derivative rule is the Substitution Rule based?

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?

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?

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)

"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