Wallis's formula

Wallis's formula states that

(1) $\displaystyle\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right)=\frac{\pi}{2}\prod_{k=2}^n\left(\frac{k-1}{k}\right)$ for even n.

(2) $\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\displaystyle\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\cdots\left(\frac{n-1}{n}\right)=\prod_{k=3}^n\left(\frac{k-1}{k}\right)$ for odd n.

Wallis's formula often works well in combination with trigonometric substitution in reducing complicated definite integrals with more manageable ones.

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Wallis's formula