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

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?

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.
• As with any special-functions answer, it's worth bearing in mind that any derivation of this result will depend on how $K_0$ is defined. (Indeed, the above integral is itself a possible definition.) – Semiclassical Jul 26 '17 at 3:35