Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 18 / b

Evaluate the recurrence boundary terms at zero and $\pi /2$. For $n>1$, both $\cos x{\sin }^{n-1}x$ and $\sin x{\cos }^{n-1}x$ vanish at both endpoints. Therefore the Wallis integrals obeyThe starting value for both is ${\int }_0^{\pi /2}1dx=\pi /2$. Thus