Concept preview

What you will be able to do

1. 01Read the STRUCTURE of an integrand — is it a known basic form, a composite times an inner derivative, a product of unlike functions, or a rational function? — and use that structure to choose an antidifferentiation technique before doing any algebra.
2. 02Apply a decision procedure that ranks the methods: try a basic antiderivative first, then $u$-substitution, and (for BC) integration by parts, partial fractions, polynomial long division, or completing the square when the structure calls for them.
3. 03Justify WHY the chosen method fits, and recognise the tell-tale sign that you have forced the wrong technique so you can back out and re-read the integrand instead of grinding forward.

Every antidifferentiation problem is the same underlying request: find a function $F$ whose derivative is the integrand $f$. Because differentiation has several rules, undoing it needs a matching menu of techniques, and the trick is to run through them in a sensible order. Here is the decision procedure, cheapest option first. (1) Basic antiderivative. Ask whether the integrand is already a standard form you can write down immediately — a power $x^n$, an exponential $e^x$, a trig function, $\frac{1}{x}$, $\frac{1}{1+x^2}$, and so on, possibly after simplifying or splitting a sum. If yes, you are done in one step. (2) $u$-substitution. If not, scan for a composite: is there an inside function $u=g(x)$ whose derivative $g'(x)$ appears (up to a constant) as a factor? If so, substitute and the integral collapses. Substitution is the reverse of the chain rule and it is the single most common non-trivial technique — always check for it before escalating.