I was wondering if anyone had come across an integral that has come up in some papers I have been reading in probability theory. I am looking for any tractable analytic expressions related to the evaluation of the integral but specifically a simple lower bound in terms of $r,u$.
Suppose $A = [a,a+1] \subset \mathbb{R^+}$ (In my case $a=u^{\frac{1}{\alpha}}$) and fix $r>1$ ($r$ is just some fractional power in the cases I deal with like $3/2$ for instance). Fix $u >0$, modulo some constants I am interested in integrals of this form.
$\int_{A} \frac{{\frac{u}{x^{r}}}}{1+\frac{u^2}{x^{2r}}}dx$
Is there any way to get a simple lower bound for this integral in terms of u or otherwise tractable analytic expression?
I think for large $u$ the behavior of the integral becomes very simple since the expression is approximate to $1/u$ but I was hoping some one had seen this in a table of integrals or there was some nice contour integral formula that I am not thinking of.