Solution
= Solution
For $x\ge1$, $(1+x)^2\le4x^2$, so $x\log x/(1+x)^2\ge(\log x)/(4x)\ge0$. The latter integral diverges because its primitive is $(\log x)^2/8$. Thus the absolute integral diverges and the answer is $\boxed{\text{not integrable on }(1,\infty)}$.