Repeated and Cyclic Integration by Parts

Repeated integration by parts is explicit AHL5.16 content. With a polynomial such as \(x^2\), differentiating repeatedly eventually gives a constant.

For a cyclic case, name the original integral \(I\). Keep the sign when applying parts for the second time; then collect the occurrences of \(I\).

\(\begin{aligned}I&=\int e^x\sin x\,\mathrm{d}x\\&=e^x\sin x-\int e^x\cos x\,\mathrm{d}x\end{aligned}\)

\(\begin{aligned}I&=e^x\sin x-e^x\cos x-I\\2I&=e^x(\sin x-\cos x)\\I&=\dfrac{e^x}{2}(\sin x-\cos x)+C\end{aligned}\)

“Cyclic” describes the return of the original integral. It does not mean an infinite integration limit.