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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.

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Is there a particular reason for writing the integrand this way, as opposed to $$\frac{ux^r}{1+ux^r}$$ or $$\frac u{u+x^{-r}}\;?$$ –  joriki Sep 16 '11 at 18:23
    
Sorry there was a typo. Thanks for pointing that out –  user7980 Sep 16 '11 at 18:28
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Note that you can get rid of $u$ by substituting $t=u^{1/r}/x$; that leaves $$\frac{x^r}{1+x^{2r}}\;,$$ in the integrand, for which Wolfram|Alpha gives a result involving the hypergeometric function, which may or may not be helpful. –  joriki Sep 16 '11 at 18:40
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@joriki: what I get with this substitution is $-u^{1/r} \int \frac{t^{r-2}}{1+t^{2r}}\ dt$. Note that there are singularities on the unit circle, so a hypergeometric series solution may converge inside or outside the circle but not both, and I'm not sure which one you want. –  Robert Israel Sep 16 '11 at 19:15
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@Robert: Sorry, I didn't type what I meant :-) I meant $1/t=u^{1/r}/x$. I didn't want to change $x$ substantially, just rescale to get rid of $u$. –  joriki Sep 16 '11 at 19:26

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