Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 13 / c

Use integration by substitution with the reciprocal variable $t=1/x$, so $dx=-dt/t^2$ and $\log x=-\log t$. The endpoints $x=1,\infty$ become $t=1,0$. ThusRenaming the dummy integration variable proves the identity without evaluating either side.

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 $x=1/t$ givesThereforeAbsolute convergence makes this cancellation between the two ends legitimate.