Evaluate expression in terms of Beta function

Question

Evaluate $\int_0^1\frac{x^{(m-1)} + x^{(n-1)}}{(1+x)^{(m+n)}}dx$ in terms of Beta function.

I am able to obtain the answer if the limits were 0 to infinity, but not with 0 to 1. Please help.

Answer 1

By taking the changing $t =\frac{1}{x}$ ie $dx = -\frac{1}{t^2}dt$

$$I_{n,m}:=\int_0^1\frac{x^{(m-1)} + x^{(n-1)}}{(1+x)^{(m+n)}}dx = \int_1^\infty\frac{x^{(m-1)} + x^{(n-1)}}{(1+x)^{(m+n)}}dx $$
so $$I_{n,m}:=\frac{1}{2}\int_0^\infty\frac{x^{(m-1)} + x^{(n-1)}}{(1+x)^{(m+n)}}dx = \frac{1}{2}\int_0^\infty\frac{ x^{(n-1)}}{(1+x)^{(m+n)}}dx +\frac{1}{2}\int_0^\infty\frac{x^{(m-1)} }{(1+x)^{(m+n)}}dx $$ finally for each terms using $u =\frac{1}{1+x}$ ie $x= 1-\frac{1}{u}$ we get $$ I_{n,m}:=\frac{1}{2}\int_0^1x^{m-1}(1-x)^{n-1}dx +\frac{1}{2}\int_0^1x^{n-1}(1-x)^{m-1}dx= B(n,m) =\frac{\Gamma(n)\Gamma(m)}{\Gamma(n+m)}$$