# Integral $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$

I'm looking to calculate $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$.

Mathematica10 can't integrate this, but numerical integration gives an answer of 5.50843 for any $a$ I've tried, so I assume that there is an analytic answer to be found.

Is this a standard integral?

Many thanks,

• Have you tried using complex analysis? – UserX Sep 22 '14 at 22:26
• Perhaps try substituting $x=a \tan{\theta}$. – Ron Gordon Sep 22 '14 at 22:39

Sub $x=a \tan{\theta}$ and the integral becomes
$$\sqrt{e} \int_{-\pi/2}^{\pi/2} d\theta \, e^{-\frac12 \cos{2 \theta}}$$
which is indeed a standard integral, equal to $\pi \sqrt{e} I_0(1/2) \approx 5.50842977...$, where $I_0$ is the modified Bessel function of the first kind of zeroth order.