Integration: the standard integrals and how to choose a method

Integration is harder than differentiation for one reason: there's no single procedure. Instead you carry a toolkit and learn to recognise which tool a given integral is asking for. Master the standard results first — most questions are one of them in disguise.

The standard integrals

The power rule is the reverse of differentiation: add one to the power, divide by the new power. Note the exception — you can't divide by zero, so $\frac{1}{x}$ integrates to a logarithm, not a power.

The three techniques

1. Reverse chain rule (substitution). Spot a function and its derivative both present. Let $u$ be the inside function and the integral simplifies. Tell-tale sign: something like $\int 2x\,(x^2+1)^5\,dx$ — the $2x$ is the derivative of $x^2+1$.
2. Integration by parts. For a product of two unlike functions (e.g. $x\,e^x$): $$\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx.$$ Choose $u$ by LIATE — Logs, Inverse trig, Algebra, Trig, Exponentials — whichever comes first becomes $u$.
3. Rewrite first. Many integrals aren't hard, just disguised. Expand brackets, split a fraction, or use a trig identity to turn $\sin^2 x$ into something you can integrate.

Don't forget what an integral is

The $\int$ sign is a stretched "S" for sum — you're adding up infinitely many thin slices. For a definite integral, evaluate the result at both limits and subtract; the $+C$ cancels and you get a number (an area).

Always add $+C$ to an indefinite integral. It's not decoration — infinitely many curves have the same gradient, and $C$ is the one mark examiners take back most often.

Last revised 18 September 2025.