Real integral: $\cos(ax)/\sqrt{1+x^2}$

Question

I have no idea to evaluate the following integral and I am in trouble:

$$\int_0^\infty\frac{\cos(ax)}{\sqrt{1+x^2}}dx$$

since the square root prevents me from using an ordinary way to evaluate real integrals with residue theorem. Does anyone know how to solve the above integral?

Thanks in advance.

Answer 1

If you have a look here, you will see that $$\int_0^\infty \frac{\cos(ax)}{\sqrt{1+x^2}}dx=K_0(|a|)$$ where appears the modified Bessel function of the second kind.