AHL 5.11 (HL)—Further integration

Use $\int x^n\,dx=x^{n+1}/(n+1)+C$ for rational $n\ne-1$, $\int x^{-1}\,dx=\ln|x|+C$, $\int\sin x\,dx=-\cos x+C$, $\int\cos x\,dx=\sin x+C$, $\int\sec^2x\,dx=\tan x+C$ and $\int e^x\,dx=e^x+C$.

For $\int f(g(x))g'(x)\,dx$, let $u=g(x)$ or recognize the reverse chain rule. Constant factors must be adjusted so the derivative of the inner function is present.

Example

$\int4x\sin(x^2)\,dx$: take $u=x^2$, so $du=2x\,dx$ and the integral becomes $2\int\sin u\,du=-2\cos(x^2)+C$.

Differentiate the result to verify it. Do not use the ordinary power rule for $x^{-1}$, omit absolute values in $\ln|x|$, or forget the inner derivative adjustment.