Integration by Substitution

Reversing the Chain Rule

Integration by substitution is the reverse of the chain rule and your most powerful tool for solving complex integrals. It transforms a hard integral into an easy one through a clever change of variable.

When your integrand contains an inner function $u = g(x)$ alongside its derivative factor, apply the formal rule:

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

Here, $u = g(x)$ and $du = g'(x)\, dx$. This collapses the composite expression into a simple integral in $u$ which you evaluate before converting back.

Worked Example & Pitfalls

Evaluate $\int 2x\cos(x^2)\, dx$. Let $u = x^2$, making $du = 2x\, dx$. Substituting these gives $\int \cos u\, du = \sin u + C = \sin(x^2) + C$.

For definite integrals, you have two valid paths:

Common Pitfalls: Never forget to convert $dx$ into $du$ by accounting for the $g'(x)$ factor. In definite integrals, never mix old $x$-limits with the new $u$-variable; change your limits immediately or substitute back first.