Show that the improper integral $\int_0^\infty \cos(x^2)$ exists but $\cos(x^2)$ is not Lebesgue integrable.
I'm asked to prove the above statement. I know that the integral is a special one, but I've not yet found a proof of its existence. And as for proving that it is not Lebesgue integrable, I don't have any idea. All tips appreciated.