Calculate $\int_{-\infty}^\infty \frac{e^{-x^2}}{x^2+a^2}\ dx$. Let $$F(a)=\int_{-\infty}^\infty \frac{e^{-x^2}}{x^2+a^2}\ dx, \quad a>0.$$ Is it possible to relate $F(a)$ to some known (special) functions? 

Parameterize this integral by adding a second parameter, $t$:
$$I(t):= \int_{-\infty}^\infty \frac{e^{-(x^2+a^2)t}}{x^2+a^2}dx$$
Differentiating with respect to $t$, we have
$$I'(t)=-\int_{-\infty}^\infty e^{-(x^2+a^2)t}dx=-\sqrt{\frac{\pi}{t}} e^{-a^2 t}$$
This shows us that
$$\begin{align}
I(t)
&=I(0)-\int_0^t \sqrt{\frac{\pi}{x}}e^{-a^2 x}dx\\
&=\frac{\pi}{a}-2\int_0^{\sqrt{t}} \sqrt{\pi}e^{-a^2 x^2}dx\\
&=\frac{\pi}{a}-\frac{\pi \text{erf}(a\sqrt{t})}{a}\\
&=\frac{\pi \text{erfc}(a\sqrt{t})}{a}\\
\end{align}$$
Which gives us the desired value of your integral:
$$\int_{-\infty}^\infty \frac{e^{-x^2}}{x^2+a^2}dx=\frac{\pi e^{a^2}\text{erfc}(a)}{a}$$
Delicious!
