integrals of exponential functions over the real axis

Question

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?

Answer 1

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...).