$x\log 2x$

Let $\int {x\log 2x\,dx}$
$= \left( {\log 2x} \right) \cdot \cfrac{{{x^2}}}{2} - \int {\cfrac{d}{{dx}}\left( {\log 2x} \right)\left( {\cfrac{{{x^2}}}{2}} \right)dx}$

$= \log \left( {2x} \right) \cdot \cfrac{{{x^2}}}{2} - \int {\cfrac{2}{{2x}}\left( {\cfrac{{{x^2}}}{2}} \right)dx} + C$
$= \cfrac{{{x^2}}}{2}\log \left( {2x} \right) - \cfrac{1}{2}\int x dx + C = \cfrac{{{x^2}}}{2}\log \left( {2x} \right) - \cfrac{1}{2} \cdot \cfrac{{{x^2}}}{2} + C$

$= \cfrac{{{x^2}}}{2}\log \left( {2x} \right) - \cfrac{{{x^2}}}{4} + C$