$x\log x$

Let $I = \int {x\log xdx} = \log x\int x dx - \int {\left[ {\cfrac{d}{{dx}}\left( {\log x} \right)\int x dx} \right]dx}$

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

$= \cfrac{{{x^2}}}{2}\log \,x - \cfrac{1}{2} \times \cfrac{{{x^2}}}{2} + C = \cfrac{{{x^2}}}{2}\log x - \cfrac{1}{4}{x^2} + C$