Hints in solving this analitically?

[Petr Mugver]

$$\int_{-\infty}^{+\infty}\frac{e^{-x^2}}{\sqrt{x^2+1}}dx$$

I calculated it numerically, but I need an exact number. Hints?

 

[Nebuchadnezza]

Well you can't express the answer purely by analyctic functions, but you could write the answer like this$$I \, = \sqrt{e} \, K_0 \left( \frac{1}{2} \right)$$

Where K_0 is the modified Bessel function of the second kind.

Or use the MejerG function to express the answer.

 

[DivisionByZro]

You can start by noticing that this is really another form of:

$$\int_{0}^{\infty }cos(x sinh (t))dt$$

Which is itself a special case of the Digamma function when v=0.

In any case, it's not easy to integrate it. You could try a series expansion centered at x=0, but it gets really messy. Your best bet is analytically. Otherwise, try starting with the Digamma function.

 

[DivisionByZro]

I was not able to edit my previous post for some reason. i meant to say that your best bet is to solve numerically, not analytically. But of course, we don't know your level of expertise with these kinds of integrals.