Fresnel integrals are not Lebesgue integrable

Question

How does one prove that the Fresnel integrals $$ S(x) = \int_0^x \! \sin(t^2) \, \mathrm{d}t, \qquad C(x) = \int_0^x \! \cos(t^2) \, \mathrm{d}t $$ do not belong to $L^1(\mathbb{R})$?

Answer 1

Since $\lim_{\lvert x \rvert \to \infty} \lvert S(x) \rvert = \sqrt{\pi/8}$, we can find $M \in \mathbb R$ such that for $\lvert x \rvert > M$, we have $$\lvert S(x) \rvert \ge \sqrt{\pi/16}.$$ But then $$\int^\infty_{-\infty} \lvert S(x) \rvert dx \ge \int^M_{-M} \lvert S(x) \rvert dx + \int^\infty_M \sqrt{\pi/16} \,\,dx + \int^{-M}_{-\infty} \sqrt{\pi/16}\,\, dx = \infty$$ and similarly for $C(x)$.