Concept preview

What you will be able to do

1. 01Recognise the pattern $\displaystyle\int f\left(g(x)\right)\,g'(x)\,dx$ hiding inside an integral and reverse the chain rule by substituting $u=g(x)$, $du=g'(x)\,dx$, integrating in $u$, and back-substituting.
2. 02Adjust for a missing constant factor when the derivative of the inner function is off by a number, and evaluate indefinite integrals with a $+C$ produced by substitution.
3. 03Evaluate DEFINITE integrals by substitution using either method — changing the limits to $u$-values $g(a)$ and $g(b)$, or finishing the antiderivative in $x$ and using the original limits — and explain why keeping the old limits with a $u$-antiderivative gives the wrong number.

Recall the chain rule in the form you will need. If $F$ is an antiderivative of $f$, then $\dfrac{d}{dx}\,F\left(g(x)\right)=F'\left(g(x)\right)\,g'(x)=f\left(g(x)\right)\,g'(x)$. Read that equation from right to left instead of left to right and it becomes an integration fact: $\int f\left(g(x)\right)\,g'(x)\,dx = F\left(g(x)\right)+C.$ In words, whenever an integrand is a composite $f\left(g(x)\right)$ multiplied by the derivative $g'(x)$ of its inner function, the antiderivative is just $F$ of the inner function. Substitution is the systematic way to recognise and exploit exactly this structure without having to guess $F\left(g(x)\right)$ by inspection.