CalcRef

Techniques of Integration · Substitution Methods

u-Substitution

∫f(g(x)) g′(x) dx=∫f(u) duwhere u=g(x)\int f(g(x))\, g'(x)\, dx = \int f(u)\, du \quad \text{where } u = g(x)

The reverse of the chain rule. Identify an inner function u = g(x), compute du = g'(x) dx, and substitute.

Worked examples

Find ∫ 2x cos(x²) dx.
  1. Let u = x², du = 2x dx
  2. ∫ cos(u) du = sin(u) + C
  3. Back-substitute: sin(x²) + C

Answer: sin(x²) + C

Find ∫ eˢⁱⁿˣ cos x dx.
  1. Let u = sin x, du = cos x dx
  2. ∫ eᵘ du = eᵘ + C = e^(sin x) + C

Answer: e^(sin x) + C

Find ∫ x/(x²+1) dx.
  1. Let u = x²+1, du = 2x dx → x dx = du/2
  2. (1/2)∫ 1/u du = (1/2) ln|u| + C

Answer: (1/2) ln(x²+1) + C

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