Antiderivative of the logarithm by parts

Integrate $\log x$ with respect to $x$ (natural logarithm, $x>0$).

$x\log x-x+C$. Integrate by parts with $u=\log x$, $dv=dx$: $\int\log x\,dx=x\log x-\int x\cdot\tfrac1x\,dx$.