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5.20 Wallis’s integral

Example

To evaluate $I_{n}=\int_{0}^{\pi/2}\cos^{n}x\,dx$ for integers $n\neq 0.$

Solution. First we find:

I_{0}=\int_{0}^{\pi/2}\cos^{0}x\,dx=\pi/2;

I_{1}=\int_{0}^{\pi/2}\cos x\,dx=\bigl[\sin x\bigr]_{0}^{\pi/2}=\sin(\pi/2)-% \sin 0=1.

We shall prove the recurrence relation

I_{n}={{n-1}\over{n}}I_{n-2}\qquad(n\geq 2);

note that we go down two steps from $n$ to $n-2$ . We integrate by parts, obtaining

Find $\int\cos^{2}x\sin x\,dx$ .