Pick a function $f(x)$ and either try to integrate its absolute value or try to integrate $\sqrt{f(x)}$ or $1/f(x)$. This will require you to solve $f(x) = 0$. And asking if there are any such solutions is undecidable for the class of functions you describe.
Edit it's not clear for a few reasons why or if this would actually do the trick.
But I feel we could pick $f(x)$ to be a polynomial without repeated roots. Then asking if the integral of $|f(x)|$ is a polynomial is undecidable.