Substitution

Level 1 - Calculus topic page in Integral Calculus.

Principle

Substitution reverses the chain rule. It is used when an integral contains an inner expression and a matching derivative factor.

\(u\)

new variable used for the inner expression

\(du\)

differential corresponding to u

\(dx\)

differential corresponding to x

Choose \(u=g(x)\), compute \(du=g'(x)dx\), replace the integral in terms of \(u\), integrate, then substitute back.

Substitution pattern

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

For definite integrals, either change the limits to \(u\)-values or substitute back before evaluating.

Question

Find \(\int 2x(x^2+1)^4\,dx\).

Answer

Let \(u=x^2+1\). Then \(du=2x\,dx\). The integral becomes \(\int u^4\,du\). Integrate: \(u^5/5+C\). Substitute back to get \(\frac{(x^2+1)^5}{5}+C\).

Question

Evaluate \(\int_0^1 2x e^{x^2}\,dx\).

Answer

Let \(u=x^2\), so \(du=2x\,dx\). When \(x=0\), \(u=0\); when \(x=1\), \(u=1\). The integral becomes \(\int_0^1 e^u\,du=e^u\big|_0^1=e-1\).