Supposing you mean #ln(x)^2=(lnx)^2#
#intxln(x)^2dx=#
#=int(x^2/2)'ln(x)^2dx=#
#=x^2/2ln(x)^2-intx^2/2(ln(x)^2)'dx=#
#=x^2/2ln(x)^2-intx^cancel(2)/cancel(2)*cancel(2)lnx*1/cancel(x)dx=#
#=x^2/2ln(x)^2-intxlnxdx=#
#=x^2/2ln(x)^2-int(x^2/2)'lnxdx=#
#=x^2/2ln(x)^2-(x^2/2lnx-intx^2/2(lnx)'dx)=#
#=x^2/2ln(x)^2-(x^2/2lnx-intx^cancel(2)/2*1/cancel(x)dx)=#
#=x^2/2ln(x)^2-(x^2/2lnx-1/2intxdx)=#
#=x^2/2ln(x)^2-(x^2/2lnx-1/2x^2/2)+c=#
#=x^2/2ln(x)^2-(x^2/2lnx-x^2/4)+c=#
#=x^2/2ln(x)^2-x^2/2lnx+x^2/4+c=#
#=x^2/2(ln(x)^2-lnx+1/2)+c#
Supposing you mean #ln(x)^2=ln(x^2)#
#intxln(x)^2dx=intx*2lnxdx#
#2intxlnxdx=#
#=2int(x^2/2)'lnxdx=#
#=2(x^2/2lnx-intx^2/2*(lnx)'dx)=#
#=2(x^2/2lnx-intx^cancel(2)/2*1/cancel(x)dx)=#
#=2(x^2/2lnx-1/2intxdx)=#
#=2(x^2/2lnx-1/2x^2/2)+c=#
#=cancel(2)*x^2/(cancel(2))(lnx-1/2)+c=#
#=x^2(lnx-1/2)+c#