For the computation of an expected value, with $m>i$ and $m,i\in\mathbb{N}$, I am trying to show that
$$ \int_0^1 x\cdot m\binom{m-1}{i-1}x^{i-1}(1-x)^{m-i}\mathrm{d}x=\frac{i}{m+1}. $$
However, I am apparently lacking some Binomial/Gamma/Factorial identities in order to simplify the sum that arises:
\begin{alignat*}{3} &\int_0^1 x\cdot m\binom{m-1}{i-1}x^{i-1}(1-x)^{m-i}\mathrm{d}x\\ =&\frac{m!}{(i-1)!(m-i)!}\int_0^1 x^i \sum_{k=0}^{m-i}\binom{m-i}{k}x^k\mathrm{d}x\\ =&\frac{m!}{(i-1)!(m-i)!}\int_0^1 (m-i)!\sum_{k=0}^{m-i}\frac{1}{k!(m-i-k)!}x^{i+k}\mathrm{d}x\\ =&\frac{m!}{(i-1)!}\sum_{k=0}^{m-i}\frac{1}{k!(m-i-k)!(i+k+1)}\\ =&\ldots \end{alignat*}
where I start being lost. I am fine using Gamma/Beta/Pochhammer notation but could not come up with identities that allowed be to simplify the form I am stuck with. Thanks in advance!
We have that $$\int_0^1 x\cdot m\binom{m-1}{i-1}x^{i-1}(1-x)^{m-i}\mathrm{d}x= i\binom{m}{i}\int_0^1 x^{i}(1-x)^{m-i}\mathrm{d}x=i\binom{m}{i}B(i+1,m-i+1)=\frac{i}{m+1} $$ where we used the definition of Beta function and the following known property: for any positive integers $x$ and $y$ then $$B(x,y)=\frac{(x-1)!(y-1)!}{(x+y-1)!}.$$