By integrating round the contour $C_{R}$, which is the boundary of the domain

$$D_{R}=\left\{z=r e^{i \theta}: 0<r<R, \quad 0<\theta<\frac{\pi}{4}\right\}$$

evaluate each of the integrals

$$\int_{0}^{\infty} \sin x^{2} d x, \quad \int_{0}^{\infty} \cos x^{2} d x$$

[You may use the relations $\int_{0}^{\infty} e^{-r^{2}} d r=\frac{\sqrt{\pi}}{2}$ and $\sin t \geq \frac{2}{\pi} t$ for $\left.0 \leq t \leq \frac{\pi}{2} \cdot\right]$