This is of course Integration by Parts.
Let:
#u = ln^3x#
#du = (3ln^2x)/xdx#
#dv = xdx#
#v = x^2/2#
#uv - intvdu#
#= x^2/2ln^3x - int x^cancel(2)/2 * (3ln^2x)/cancel(x)dx#
#= (x^2ln^3x)/2 - 3/2int xln^2xdx#
Repeat:
#u = ln^2x#
#du = (2lnx)/xdx#
#dv = xdx#
#v = x^2/2#
#= (x^2ln^3x)/2 - 3/2(int xln^2xdx)#
#= (x^2ln^3x)/2 - 3/2((x^2ln^2x)/2 - int x^cancel(2)/cancel(2) * (cancel(2)lnx)/cancel(x) dx)#
#= (x^2ln^3x)/2 - 3/2((x^2ln^2x)/2 - int xlnx dx)#
and repeat again:
#u = lnx#
#du = 1/xdx#
#dv = xdx#
#v = x^2/2#
#= (x^2ln^3x)/2 - 3/4x^2ln^2x + 3/2(int xlnx dx)#
#= (x^2ln^3x)/2 - 3/4x^2ln^2x + 3/2((x^2lnx)/2 - int x^cancel(2)/2*1/cancel(x)dx)#
#= (x^2ln^3x)/2 - 3/4x^2ln^2x + 3/2((x^2lnx)/2 - int x/2dx)#
#= 1/2x^2ln^3x - 3/4x^2ln^2x + 3/4x^2lnx - 3/4int xdx#
#= 1/2x^2ln^3x - 3/4x^2ln^2x + 3/4x^2lnx - 3/8x^2#
#= color(blue)(1/8x^2(4ln^3x - 6ln^2x + 6lnx - 3) + C)#