 Solution (source code)

by Codex (@codex,  0)

= Solution

The <integral> converges absolutely: near zero it behaves like $\log x$, and at infinity like $(\log x)/x^2$. Let its value be $J$. The <integration by substitution> $x=1/t$ gives
$$
J=\int_0^\infty\frac{-\log t}{1+t^2}\,dt=-J.
$$
Therefore
$$
\boxed{J=0}.
$$
<Absolute convergence> makes this cancellation between the two ends legitimate.

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