TITLE: Evaluate $\int_0^\infty\frac{\ln x}{1+x^2}dx$
QUESTION [47 upvotes]: Evaluate $$\int_0^\infty\frac{\ln x}{1+x^2}\ dx$$
I don't know where to start with this so either the full evaluation or any hints or pushes in the right direction would be appreciated.  Thanks.

REPLY [19 votes]: Here's another very short solution.
The substitution $x \to e^t$ transforms the integral to
$$\int_{-\infty}^{\infty} \frac{t}{e^t+e^{-t}}\,dt$$
which is zero by the anti-symmetry of the integrand.