$$\int_0^\infty{\ln x}\mathrm dx$$ is the integral in question.
Taking the integral, I am not quite sure what to do with the 0, as ln is not defined there. Should I evaluate from 0+?
A related question: what is the proper term that describes improper integrals that have undefined values on both bounds?
Rewrite it as $$\lim_{\epsilon \to 0^+}\int_{\epsilon}^{1}\ln x\mathrm dx+\lim_{\delta\to \infty}\int_{1}^{\delta}\ln x\mathrm dx$$
Either way this improper integral is divergent.