Integration by substitution

Not every integral has an obvious antiderivative. u-substitution is the first and most important technique for the ones that don't — and it's simply the chain rule run backwards.

Undoing the chain rule

The chain rule turned f(g(x)) into f′(g(x))·g′(x). So when you spot an integral that looks like a function of something, times the derivative of that something, you can reverse it. The move: rename the inner function u.

Set u = g(x), so that du = g′(x) dx. Substituting turns a tangled integral in x into a clean one in u.

The recipe

1. Choose u — usually the inner function, or whatever's under a power, root, or exponent.
2. Compute du = g′(x) dx, and match it to what's in the integral.
3. Rewrite the whole integral in terms of u (no x may remain).
4. Integrate in u, then substitute back x.

Worked example

Evaluate ∫ 2x·(x² + 1)³ dx.

• Choose u = x² + 1. Then du = 2x dx — and a "2x dx" is sitting right there.
• Rewrite: ∫ u³ du.
• Integrate: u⁴/4 + C.
• Substitute back: (x² + 1)⁴ / 4 + C.

Differentiate that answer and the chain rule hands you 2x·(x² + 1)³ — right back where you started.

A note on definite integrals

For a definite integral, either substitute x back before plugging in the original limits, or — cleaner — convert the limits to u and evaluate directly. If x goes from 0 to 1, then u = x² + 1 goes from 1 to 2, and you never return to x.

Why this matters

u-substitution is the reverse-chain-rule that unlocks a huge share of the integrals you meet — and it's the foundation the fancier techniques (integration by parts, trig substitution) build on. Master spotting the "inner function and its derivative" pattern and most integrals open up.