To solve
    ∫x ln x dx
(ignoring the +c since we are going to use this in working out 
the answer over a given span> we could set u=ln x and dv=x dx.

That would make our du=1/x dx and our v=(1/2)x2.

So, since ∫u dv = uv - ∫v du, we would get
   (1/2)x2ln x - ∫(1/2)x2*(1/x)dx
which would simplify to
   (1/2)x2ln x - (1/2)∫x dx
which works out to
   (1/2)x2ln x - (1/2)*(1/2)x2
which would simplify to
   (1/4)x2 (2 ln x - 1)