Integration by parts formula is ∫u⋅dv=u⋅v−∫v⋅du
From that u=x, du=dx
dv=logx, ∫ logx dx=x(log(x)-1)+C
∫ logx dx= xlog(x)-x+C
∫ x lnx dx= (½)x2 log x - (x2/4) + C