Is there any general closed form to the following? $$\int^1_0 f^2(x)F^{n-1}(x)dx$$ when F is a CDF with support $[0,1]$ and $f$ is its corresponding pdf.
It's easy when it is $\int^1_0 f(x)F^{n-1}(x)dx$, but what happens when the pdf is powered by some number?