The Substitution Rule

The substitution rule is simply a way to make a messy integral simple by renaming part of it. All we're really doing is giving the "inside" of a complicated expression a short name, $u$, so the integral looks like one we already know. The formula has three pieces: $f(g(x))$ (a function with something inside it), $g'(x)\,dx$ (the derivative of that inside part, sitting next to it) and $f(u)\,du$ (what's left after renaming $g(x)$ as $u$). It looks heavy, but it's just a renaming.

Take a look at the chain rule from the derivatives unit: $\frac{d}{dx}F(g(x)) = F'(g(x))\,g'(x)$. Notice the shape of the answer: a function of $g(x)$, multiplied by $g'(x)$. So when an integral has exactly that shape, we know it came from a chain rule, and the antiderivative is $F(g(x))$. Substitution is a tidy way to see that shape: set $u = g(x)$, and the $g'(x)\,dx$ turns into $du$.

Steps:

1. Choose $u$, usually the "inside" function.
2. Compute $du = u'\,dx$.
3. Rewrite the whole integral in terms of $u$. No $x$ may remain.
4. Integrate, then substitute $u = g(x)$ back.

Constant factors are fine. If $du = 3x^2\,dx$ but you only have $x^2\,dx$, use $x^2\,dx = \frac13\,du$.

A common mistake. It's tempting to fix a missing $x$ the same way, dividing by it as if it were a constant. It doesn't work: $x$ changes as you integrate, so it can't be pulled outside the integral. Factors containing $x$ can't be fudged; if they don't match, try a different $u$.

Check by differentiating: the chain rule should give back the original integrand.

Substitution is a change of units: measuring in $u$ instead of $x$. Physicists do it constantly, switching from time to angle, or from position to energy, to make an integral simple.