Answer
$$-\frac{\sqrt 2+2}{4}.$$
Work Step by Step
Since $(\sin(2x)'=2\cos(2x)$, we have
$$\int_{\pi/4 }^{5 \pi/8 } \cos(2x)d x= \frac{1}{2}\sin(2x) |_{\pi/4 }^{5 \pi/8 }\\
= \frac{1}{2}\sin(5\pi/4)-\frac{1}{2}\sin(\pi/2)=-\frac{\sqrt 2}{4}- \frac{1}{2} \\
=-\frac{\sqrt 2+2}{4}.$$
