$ \int_{0}^{1}x^m.(1-x)^{15-m}dx$ where $m\in \mathbb{N}$ $=\displaystyle \int_{0}^{1}x^m.(1-x)^{15-m}dx$ where $m\in \mathbb{N}$
My Try:: Put $x=\sin^2 \theta$ and $dx = 2\sin \theta.\cos \theta.d\theta$ and changing limit, We Get
$ = \displaystyle \int_{0}^{\frac{\pi}{2}}\sin^{2m}\theta.\cos^{30-2m}\theta.2\sin \theta.\cos \theta d\theta$
$ = \displaystyle 2\int_{0}^{\frac{\pi}{2}} \sin^{2m+1}\theta.\cos^{31-2m}\theta d\theta$
Now How can i proceed after that.
Is there is any method to Calculate the Given Integral Then plz explain here.
Thanks

The $\beta$-function is defined as $$\beta(m,n) = \int_0^1 x^{m-1}(1-x)^{n-1} dx = \dfrac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$$ In your case, $$\int_0^1 x^{m+1-1}(1-x)^{16-m-1} dx = \beta(m+1,16-m) = \dfrac{\Gamma(m+1) \Gamma(16-m)}{\Gamma(17)} = \dfrac{m! (15-m)!}{16!}$$
