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?

2–74. 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

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?

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