Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 17 / c / Solution

First apply integration by parts with $u=x^k$ and $dv=\sin xdx$:For $k>1$, integrate the remaining integral by parts once more. Its boundary term $[x^{k-1}\sin x{]}_0^{\pi }$ is zero, soAt $k=1$, direct integration gives $F(1)=\pi$, agreeing with the same formula; the coefficient of $F(-1)$ vanishes and that integral is finite. Now follow the odd chain:This is the sine-weighted power integral recurrence.