Repeated integration by parts is explicit AHL5.16 content. With a polynomial such as x2, 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. I= exsin x,dx =exsin x- excos x,dx I=exsin x-excos x-I 2I=ex(sin x-cos x) I= ex2(sin x-cos x)+C “Cyclic” describes the return of the original integral. It does not mean an infinite integration limit.

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.

Techniques of Integration

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Repeated and Cyclic Integration by Parts

Repeated and Cyclic Integration by Parts