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