Does $\int_{e}^{\infty} \frac{1}{x+\ln(x)} \, dx$ diverge or converge?

Question

I'm not sure if this should be solved using comparison test or absolute integration. How can I tell if $\int_{e}^{\infty} \frac{1}{x+\ln(x)} \, dx$ diverges or converges?

Answer 1

since $x> \ln (x)>0 \ \forall x>1 $ so that $\frac{1}{x+ \ln(x)}<\frac{1}{2x}$

then $\int_e ^ \infty \frac{dx}{2x}<\int_e ^ \infty \frac{dx}{x+\ln(x)}$ and $\int_e ^ \infty \frac{dx}{2x} = \frac{1}{2}\left(\lim\limits_{b \to \infty }\ln(b) -1 \right)\to \infty $ so it diverge then $\int_e ^ \infty \frac{dx}{x+\ln(x)}$ also diverge