$\displaystyle

\int_{0}^{\infty} \cos (x^{2}) dx = \int_{0}^{\infty} \sin (x^{2}) dx = \frac{1}{2}\sqrt{\frac{\pi}{2}}

$

Can this be proved using Gamma functions without using complex numbers or De Moivre's Theorem? I'm doing problems on Gamma and Beta functions yet and I have this problem in my exercise and I haven't covered extension of integrals to complex numbers yet so I'm assuming that you can do this without using complex numbers but I fail to understand how!