Let $f:(0,1) \times \mathbb{R}$, by $f(u,x)= \frac{e^{ux}}{e^x+1}$ and $m \in \mathbb{N}_0$. Show that:
$x \mapsto x^m f(u,x)$ is for $u \in (0,1)$ in $\mathbb{R} = (-\infty,0]\cup[0,\infty)$ lebesgue-integrable.
I need some help for this exercise. I don't how to start... Would be the frist step to show that $f(u,x)= \frac{e^{ux}}{e^x+1}$ is lebesgue-integrable or should I begin with $x^m f(u,x)$?
And which theorems would be helpful to have a better understanding of the problem?
I appreciate any Hint and explanations I get.