Per the title, do the integrals $\displaystyle\int_0^\infty \frac{\cos(\ln(x))}{x}\,dx$ and/or $\displaystyle\int_0^{\pi/2} \frac{\ln(\sin x)}{\sqrt{x}}\,dx$ converge?
Attempt
I've no idea how to approach this. Dirchlet test doesn't tell me they converge/diverge, and the functions aren't nonnegative so I'm not sure if I can use the comparison test...