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
