Recipe

Integrate by u-substitution

Use when: the integrand looks like $f(g(x))\cdot g'(x)$ -- an outer function wrapped around an inner one, times something that could be that inner function's derivative

Goal: an antiderivative in terms of $x$ , found by temporarily rewriting the whole integral in terms of $u$

Taught in Choosing u: recognizing a substitution · a later section extends this recipe with more cases

Quick recipe

Choose $u$
Compute $du$
Check the leftover factor against $du$
Rewrite the entire integral in terms of $u$
Integrate in $u$ , then substitute back

1
Let $u$ be the inner function -- usually whatever's inside a radical, a trig function, an exponent, or raised to a power.

Example

For $\int 2x\cos(x^2)\,dx$ : let $u=x^2$ .

Why: the inner function is exactly the piece whose derivative the chain rule would have produced if this integral came from differentiating something -- u-substitution is that process in reverse.
2
Differentiate $u$ with respect to $x$ , then treat $du$ and $dx$ as if they could be separated.

Example

$u=x^2$ gives $du=2x\,dx$ .

Why: $du$ is what tells you whether the "leftover" part of the integrand actually matches (up to a constant) what's left in the integral once $u$ is substituted in.
3
Confirm the rest of the integrand is a constant multiple of $du$ .
- it matches exactly (or by a constant factor) — continue to Step 4.
- it doesn't match at all — this choice of $u$ doesn't work; try a different inner function.
Why: if the leftover doesn't correspond to $du$ , the integral can't be fully rewritten in terms of $u$ alone, and the substitution stalls halfway through.
4
Replace every occurrence of $x$ (including inside $dx$ ) so the integral contains only $u$ and $du$ .

Example

$\int 2x\cos(x^2)\,dx=\int\cos(u)\,du$ .

Why: this is the actual payoff of Steps 1–3 — a integral that may have looked unfamiliar in $x$ often becomes one of the basic antiderivative forms once written purely in $u$ .
5
Find the antiderivative in terms of $u$ , then replace $u$ with its original expression in $x$ .

Example

$\int\cos(u)\,du=\sin(u)+C=\sin(x^2)+C$ .

Why: the original problem asked for an antiderivative in terms of $x$ . $u$ was only ever a bookkeeping device to get there.

Common mistakes

Forgetting to substitute back to $x$ at the very end, leaving $u$ in the final answer.
Choosing a $u$ whose leftover factor doesn't actually match $du$ , and forcing the substitution anyway.
Dropping a constant factor between the leftover and $du$ instead of carrying it through.
Substituting $u$ into part of the integrand but leaving $dx$ unconverted.