Use integration by parts with .
integrate by parts y^2(log(y)) between 1 and x thats int,y^2(multiplie by logy) between 1 and x in the form A(x)+{intB(y) between 1 and xwith respect to y},,,,,,,,,,,,the answer is ((x^3)/3)multiplied by(logx)-((y^2)/2),,,,,,,sorry i dont quite have latex prrammed in my brain at the moment i will in the next few weeks,,if somebody has the method for this it wuld be much appreciated within the next 10 hours if not after,,,thanks and sorry for the inconvenience of presentation.