Summing a series of integrals I asked this question on Mathoverflow, but it was off-topic there (though it is related to my research...) and I was told to ask it here.
I have a series of integrals I would like to sum, but I don't understand how I would begin to do that considering the structure of the integrals.
Question: How do I sum (or at the very least approximate the sum of) these integrals?
Question 2:Is this question deceptively difficult, or actually difficult? 
The sum is written as follows:
$$\sum_{i=m}^n \left(\int_{0}^i\frac{i}{(i+x^2)^\frac{3}{2}} \, \mathrm{d}x \right)^2$$
$ $
where $m$ and $n$ are integers $s.t.$ $n>m> 0$. This problem looks ridiculously difficult.

This really isn't so bad. 
$$\int_0^i\frac{i}{(i+x^2)^{3/2}}dx=\frac{i}{\sqrt{i(i+1)}}.$$
So you're summing:
$$\sum_{i=m}^n\frac{i}{i+1}.$$
The latter sum unfortunately doesn't have an explicit form, unless you're willing to use digamma functions.
