integrals of exponential functions over the real axis How to evaluate the integral
$$
\int_{-\infty}^\infty \exp(-\sqrt{1+x^2})dx?
$$
I intend to change the variable $x$ to $\tan t$ but failed... How to solve it?

If you make the change of variable $u=\sqrt{1+x^2}$ you get
$$
\begin{align}
\int_{-\infty}^\infty \exp(-\sqrt{1+x^2})\:dx&=2\int_0^\infty \exp(-\sqrt{1+x^2})\:dx\\\\
&=2\int_1^\infty \frac{u}{\sqrt{u^2-1}}e^{-u}\:du
\end{align}
$$ and the latter integral is a representation of the modified Bessel function of the second kind $( eq.(7))$ giving

$$
\int_{-\infty}^\infty \exp(-\sqrt{1+x^2})\:dx=2K_1(1).
$$

You may find many properties of that function here (power series expansions, asymptotics, differential equations, approximations, generating function, integral representations...).
