Yes, I got it! The integral that I got is -1/2 but how come the answer that my professor gave is ∞? Is there any explanation to that?
Did you try doing the integration in the opposite order yet? (y first, then x). I suspect that your professor was not referring to the integral you just performed, but rather the integral of the absolute value of the integrand. To deduce that the integral of |(x-y)/(x+y)3| is infinity, you should do the integral you just solved in the order I suggested and understand what that result means.