AHL 5.16—Integration by substitution, parts and repeated parts

Integration by substitution (sometimes called u-substitution) is the integration counterpart of the chain rule. It transforms a complicated integral into a simpler one by replacing a portion of the integrand with a new variable.

The core idea: if your integral has the structure

$\int kf(g(x))g^{\prime }(x)dx$

you can let $u=g(x)$, so that $du=g^{\prime }(x)dx$. The integral then becomes:

$k\int f(u)du$

which , if you chose $u$ wisely , is much easier to evaluate.

Integration by Substitution: A technique that simplifies an integral by replacing a composite expression $g(x)$ with a new variable $u$, converting $\int f(g(x))g^{\prime }(x)dx$ into $\int f(u)du$.

The method only helps if the resulting integral in $u$ is simpler. Always think before choosing your substitution: will this actually make things easier?

Follow these five steps systematically:

1. Identify a suitable substitution $u=g(x)$ , look for an inner function whose derivative also appears in the integrand.
2. Differentiate to find $\frac{du}{dx}=g^{\prime }(x)$, then rearrange to express $dx$ (or $g^{\prime }(x)dx$) in terms of $du$.
3. Rewrite the entire integral in terms of $u$ only , no $x$ should remain.
4. Integrate with respect to $u$.
5. Back-substitute to express the answer in terms of $x$.