Integration by Substitution

Reversing the Chain Rule

Just as differentiation has the chain rule for compositions, integration has substitution — the reverse of the chain rule, and the most-used integration technique. It transforms a hard integral into an easy one by a clever change of variable.

The idea: if the integrand contains an "inner function" $u = g(x)$ together with (a multiple of) its derivative $g'(x)$, substitute: $$\int f(g(x))\,g'(x)\, dx = \int f(u)\, du, \qquad u = g(x),\ du = g'(x)\, dx.$$ Replacing $g(x)$ by $u$ and $g'(x)\,dx$ by $du$ collapses the composite into a simple integral in $u$, which you evaluate and then convert back. For example $\int 2x\cos(x^2)\, dx$: let $u = x^2$, so $du = 2x\, dx$, giving $\int \cos u\, du = \sin u + C = \sin(x^2) + C$.

The recipe:

1. Choose $u$ — usually the inner function whose derivative also appears (up to a constant).
2. Compute $du = g'(x)\, dx$ and rewrite the whole integral in terms of $u$.
3. Integrate in $u$, then substitute back $u = g(x)$ (for an indefinite integral).

For a definite integral, either substitute back and use the original limits, or — cleaner — change the limits to $u$-values ($x = a \Rightarrow u = g(a)$) and evaluate directly in $u$, never converting back.

Common pitfall: substituting for $u$ but forgetting to convert $dx$ into $du$ (i.e. neglecting the $g'(x)\,dx = du$ factor), and, in a definite integral, using the old $x$-limits with the new $u$-variable. Every $x$ (including the differential) must become $u$: if you change the variable, change the limits too — or substitute back before applying them. Leaving a stray $dx$ or mismatched limits is the defining error of substitution.