I just started learning about improper integrals. Many of them are improper because the function evaluates to infinity at some point in their domains, e.g. $f(x)=1/x$ on the domain of $(0,1)$. My question is :

Is there any function integrable on $(0,1)$ with $f\rightarrow \infty$ as $x\rightarrow 0^+$?

The answer might be no, because $f$ is unbounded. But I am not sure.