It is known that $$\int^1_0t^k(1-t)^{n-k}dt=\frac{k!(n-k)!}{(n+1)!}.$$ We can get this result using beta function or incomplete beta function. My question is what would be the value of the following: $$\int^1_0f(t)^k(1-f(t))^{n-k}dt$$ for some strictly increasing $f$ with $f(0)=0$ and $f(1)=1$? Is there any expression related to the value? I also wonder whether there's any expression for the following form $$\int^1_0t~f(t)^k(1-f(t))^{n-k}dt.$$
Any suggestion or hint to find it will be highly appreciated.
Define $u=f(t)$ so $du=f' (t) dt$ and $dt=\frac{du}{f'f^{-1}(u)}$. The integrals you've requested are the $p\in\{ 0,\,1\}$ cases of $\int_0^1 \dfrac{(f^{-1}(u)))^p u^k(1-u)^{n-k}}{f'f^{-1}(u)}du$. Given how variable the function $f'f^{-1}$ can be, I doubt some more interesting expression is obtainable in general.