Integrability of $\frac{x \log(x)}{(1+x)^{2}}$ on $(1, \infty)$

Question

Question: Is $\frac{x \log(x)}{(1+x)^{2}}$ Lebesgue-integrable on $(1, \infty)$?

I am not sure how to approach this problem, any help would be appreciated.

Answer 1

Hint

Since $x\geq 1$, $$\frac{x\log(x)}{(1+x)^2}\geq \frac{\log(x)}{4x}.$$