Simplify $\int_0^1 \int_0^x F(x,y)\, dy \,dx$ to 1-dim. integral

Question

$F(x,y)$ can be written as $f(y)$, where $f$ is integrable.

I mean I can just define me a function $\displaystyle g(x) = \int_0^x f(y)$ that only depends on $x$ and then integrate it over $[0,1]$, but I wonder if there is a better solution.