Reversing the chain rule

Differentiating a composite function is done with the chain rule: the derivative of $F(u(x))$ is $F'(u(x))\,u'(x)$. Substitution (change of variable) is nothing more than this rule read backwards, in order to integrate.

The idea: when an integral contains a composite function $f(u(x))$ multiplied by the derivative $u'(x)$ of the inner function, we can "rename" $u(x)$ as a simple letter $u$.

$$\int f(u(x))\,u'(x)\,\mathrm{d}x = \int f(u)\,\mathrm{d}u$$

All the work consists in spotting the pair $(u, u')$ in the integrand: an inner function $u$, and its derivative $u'$ present as a factor.

In the integrand	We set	And then
$f(u(x))\,u'(x)$	$u = u(x)$	$\mathrm{d}u = u'(x)\,\mathrm{d}x$

It is the derivative that "supplies" the $\mathrm{d}u$.

An illuminating example: the integral of $2x\,e^{x^2}$. Here the inner function is $u = x^2$, and its derivative $u' = 2x$ is precisely present as a factor. That is the perfect signal for a substitution.

$$u = x^2 \quad \Longrightarrow \quad \mathrm{d}u = 2x\,\mathrm{d}x \qquad\text{so}\qquad \int 2x\,e^{x^2}\,\mathrm{d}x = \int e^u\,\mathrm{d}u$$

The complicated integral turns into an elementary one. The difficulty is never the computation itself: it is spotting the right $u$, the one whose derivative is already there. Once you have this reflex, a huge family of integrals becomes accessible.