Fix an integer $n\ge 2$ and let $f: \mathbf{R}\to [0,\infty)$ be a density function, that is, a continuous function such that $\int_{\mathbf{R}}f(x)\mathrm{d}x=1$.
In addition, for each $x \in \mathbf{R}$, set $F(x):=\int_{-\infty}^x f(t)\mathrm{d}t$.
Question. Is it possible to evaluate / provide a "sufficiently good approximation" of the integral $$ \int_{\mathbf{R}} F^n(ax)f(x)\mathrm{d}x, $$ where $a$ is a fixed positive constant?