Answer
$$\frac{1}{2}-\frac{1}{\sqrt{2}}$$
Work Step by Step
\begin{align*}
\int_{0}^{\pi / 4} \int_{\pi / 4}^{\pi / 2} \cos (2 x+y) d y d x&=\int_{0}^{\pi / 4} \sin (2 x+y) \bigg|_{\pi / 4}^{\pi / 2} d x\\
&= \int_{0}^{\pi / 4} \sin (2 x+\frac{\pi}{2})- \sin (2 x+\frac{\pi}{4})d x\\
&=\frac{-1}{2}\cos (2x+\pi/2)+\frac{1}{2}\cos (2x+\pi/4)\bigg|_{0}^{\pi/4}\\
&=\frac{1}{2}-\frac{1}{\sqrt{2}}
\end{align*}
