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Pattern – 35 : Definite integration

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⋆ Pattern – 35

I_{m,n}= \int_{0}^{\frac{\pi}{2}}{\sin^mx\ \times \cos^nx\ \ dx}

= \begin{cases} & \frac{m-1}{m+n}\times \frac{m-3}{m+n\ -2}\times \frac{m-5}{m+n\ -4}\times ... \times I_{0,n} \text{where m is odd} \\ & \frac{m-1}{m+n}\times \frac{m-3}{m+n\ -2}\times \frac{m-5}{m+n\ -4}\times ... \times I_{1,n} \text{where m is even} \end{cases}

Where, I_{0,n} = \int_{0}^{\frac{\pi}{2}}{ \cos^n x\ \ dx}   and  I_{1,n} = \frac{1}{n\ +\ 1}

Proof :

We have,

I_{m,\ n}=\ -\left[\frac{\sin^{m-1}{x}\cos^{n+1\ }{x}}{m+n}\right]_0^{\frac{\pi}{2}}+\ \frac{m-1}{m+n}\ I_{m-2,n}

=\ \frac{m-1}{m+n}\times I_{m-2,\ n}

=\ \frac{m-1}{m+n}\times\ \frac{m-3}{m+n-2}\times I_{m-4,\ n}

=\ \frac{m-1}{m+n}\times\frac{m-3}{m+n-2}\times\frac{m-5}{m+n-4}\times I_{m-6,\ n}

= \begin{cases} & \frac{m-1}{m+n}\times \frac{m-3}{m+n\ -2}\times \frac{m-5}{m+n\ -4}\times ... \times I_{0,n} \text{where m is odd} \\ & \frac{m-1}{m+n}\times \frac{m-3}{m+n\ -2}\times \frac{m-5}{m+n\ -4}\times ... \times I_{1,n} \text{where m is even} \end{cases}

I_{0,n}=\int_{0}^{\pi/2}\sin^0{x}\times\cos^n{x}\ dx

=\ \int_{0}^{\pi/2}\cos^n{x}dx

I_1,n=-\left[\frac{{cos}^{n+1}{x}}{n+1}\right]_0^{\frac{\pi}{2}}

=-\frac{1}{n+1} \left[0-1\right]

=\frac{1}{n+1}

 

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