Summing a series of integrals

Question

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.

Answer 1

Because $i$ is a constant with respect to the variable of integration, we can use the fact that for any constant $b$:

$$\int \frac{b}{(b+x^2)^{3/2}}dx = \frac{x}{\sqrt{x^2 + b}}$$

so that

$$\int_0^i \frac{i}{(i+x^2)^{3/2}dx} = \frac{i}{\sqrt{i^2 + i}} = \sqrt{\frac{i^2}{i^2+1}}.$$

When we square this (to get the $i$th term in the sum), we get:

$$\left(\int_0^i \frac{i}{(i+x^2)^{3/2}dx}\right)^2 = \left(\frac{i}{\sqrt{i^2 + i}}\right)^2 = \frac{i^2}{i^2+i} = \frac{i}{i+1}$$

So, overall, your sum is simply:

$$\sum_{i=m}^n \frac{i}{i+1}$$

Answer 2

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.