Use integration by parts to integrate ∫ xlnx dx

∫ u(dv/dx) dx = uv − ∫ v(du /dx)dx is the Integration by Parts formula. 

If you set u=lnx, differentiation (rememeber from tables) leads to du/dx= 1/x, and dv/dx=x and so v=x^2/2 (raise power by one then divide by that).

Plugging this into the equation, f(x)=(x^2/2)lnx- ∫(x^2/2)/x dx, just taking the RHS integral -> 1/2∫x dx = x^2/4 +C and so combining all of this f(x)=(x^2/2)lnx-x^2/4 +C.