Example for integrable function $f(x,y)$ such that $f_x$ is not integrable

Question

Is there an example for integrable function $f(x,y)$ such that $f_x$ is not integrable. Here $f_x(y)=f(x,y)$ Is a function of only one variable $y$, means that $x$ Is a constant.

I know that from Fubini's theorem, $f_x$ cannot be integrable only on a set of measure $0$.

So is such an example exists?