\(\int_{\frac{\pi}{12}} ^{\frac{\pi}{4}} 2 \cos 2x \mathrm {d} x\) = \([\frac{2 \sin 2x}{2}]|_{\frac{\pi}{12}} ^{\frac{\pi}{4}}\) = \(\sin 2x |_{\frac{\pi}{12}} ^{\frac{\pi}{4}}\) = \(\sin 2(\frac{\pi}{4}) - \sin 2(\frac{\pi}{12})\) = \(\sin \frac{\pi}{2} - \sin \frac{\pi}{6}\) = \(1 - \frac{1}{2} = \frac{1}{2}\)