What is walli’s formula in integration ?

Walli’s Formula :

If m,n \(\in\) N & m, n \(\ge\) 2, then

(a)  \(\int_{0}^{\pi/2}\) \(sin^nx\)dx = \(\int_{0}^{\pi/2}\) \(cos^nx\)dx = \((n-1)(n-3)….(1 or 2)\over {n(n-2)….(1 or 2)}\) K

where K = \(\begin{cases} \pi/2 & \text{if n is even}\ \\ 1 & \text{if n is odd}\ \end{cases}\)

(b)  \(sin^nx.cos^mx\)dx = \([(n-1)(n-3)….(1 or 2)][(m-1)(m-3)….(1 or 2)]\over {(m+n)(m+n-2)(m+n-4)….(1 or 2)}\) K

where K = \(\begin{cases} \pi/2 & \text{if both m and n are even}\ \\ 1 & \text{otherwise}\ \end{cases}\).

Example : Evaluate : \(\int_{-\pi/2}^{\pi/2}\) \(sin^4x cos^6x\)dx

Solution : We have,

I = \(\int_{-\pi/2}^{\pi/2}\) \(sin^4x cos^6x\)dx = 2 \(\int_{0}^{\pi/2}\) \(sin^4x cos^6x\)dx 

I = 2\((3.1)(5.3.1)\over 10.8.6.4.2\) \(\pi\over 2\) = \(3\pi\over 6\)

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