 Solution

Use $u=1/x$, so $dx=-du/u^2$ and $\ln x=-\ln u$. Reversing the integration bounds gives, without evaluating either side,For $J={\int }_0^{\infty }\ln x/(1+x^2)dx$, the same change of variables givesso $\boxed{J=0}$. These are legitimate transformations of improper integrals: $|\ln x|$ is integrable near zero, and $(\ln x)/x^2$ is integrable at infinity. In particular $J$ is absolutely convergent, so no cancellation of divergent endpoints is being assumed.