$11\left(\frac{\pi}{6}, \frac{\pi}{2}\right),\left(\frac{5 \pi}{6}, \frac{3 \pi}{2}\right)$ Hint: Differentiating the function $y=\cos 2 x+2 \sin x$ yields
$$
\begin{aligned}
y^{\prime} & =-2 \sin 2 x+2 \cos x \\
& =-2 \cos x(2 \sin x-1) .
\end{aligned}
$$

Let $y^{\prime}=0, \\ \cos x>0,\end{array}\right.$ or
$$
\left\{\begin{array}{l}
2 \sin x-1<0, \\
\cos x<0 .
\end{array}\right.
$$

From the unit circle, we know that the function $y=\cos 2 x+$ $2 \sin x$ is monotonically decreasing in the intervals $\left(\frac{\pi}{6}, \frac{\pi}{2}\right),\left(\frac{5 \pi}{6}, \frac{3 \pi}{2}\right)$.
(11) The function $y=\cos 2x + 2 \sin x, x \in (0, 2\pi)$ is monotonically decreasing in the interval
$\qquad$.