Lesson 5 of 14

U-Substitution

The chain rule in reverse: spotting composite structure inside integrals.

The method

U-substitution reverses the chain rule. If an integrand contains a function and (a multiple of) its derivative, substitute:

\int f(g(x))\, g'(x)\,dx = \int f(u)\,du, \quad u = g(x)

Example: $\int x e^{x^2} dx$ . Let $u = x^2$ , $du = 2x\,dx$ , so $x\,dx = \frac{du}{2}$ :

\int x e^{x^2}\,dx = \frac{1}{2}\int e^u\,du = \frac{1}{2}e^{x^2} + C

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