Fresnel integrals. Show that the following improper Riemann integrals exist:

\[\int_{0}^{\infty} \sin x^{2} d x \text { and } \int_{0}^{\infty} \cos x^{2} d x\]

Do they exist as Lebesgue integrals?

Remark. The above integrals have the value \(\frac{1}{2} \sqrt{\frac{\pi}{2}}\). This can be proved by methods from complex analysis: Cauchy's theorem or the residue theorem.