##x(lnx)^2 -2xlnx +2x+C##

To integrate ##(lnx)^2##, let ##x= e^y## so that ##dx= e^y dy##

##int (lnx)^2 dx= int y^2 e^ydy##. Now integrate by parts,

##y^2 e^y -int 2ye^y dy##. Now again integrate by parts,

##y^2 e^y -2[ ye^y- int e^ydy]##

##y^2e^y -2ye^y +2e^y## +C

##x(lnx)^2 -2xlnx +2x+C##