Proof of integral Is there an analytical method to show that $$ \int_{-a}^a\exp\left(\frac{-1}{1-(x/a)^2}\right)\,\mathrm{d}x=ka, $$ for $a>0$. I have confirmed this result numerically for a range of values of $a$. This numerical investigation gave $k\approx0.439938161681\pm5\times10^{-13}$. This integral arose while attempting to approximate a function. I originally planned to just evaluate this integral numerically until I came across the above relationship. Applying Leibniz rule for differentiation w.r.t. $a$ under the integral didn't lead to any obvious simplifications. Is there a technique that can be used to confirm this relationship? Or is it just a coincidence?

If you are just looking to confirm the relationship, then the problem is not difficult. If, however, you are looking for a closed, exact, form for $k$, I don't know what to tell you.

First observe that

$$ \int_{-1}^{1} \exp\left( \frac{-1}{1 - x^2} \right) dx = k $$

is well-defined (ie: the integrand is integrable). Now do a change of variables, sending $x$ to $x/a$ to get:

$$ \int_{-a}^{a} \exp\left( \frac{-1}{1 - (x/a)^2} \right) \frac{dx}{a} = k. $$

This is the relationship you have above (after bringing $a$ to the RHS of course). So this relationship is by no means a coincidence, it's follows from that fact that the integral is just being scalled by a factor of $a$ and hence the answer is being scaled by a factor of $a$.