11. The function
$$
y=\cos 2x + 2 \sin x \quad (x \in (0, 2\pi))
$$

is monotonically decreasing in the interval $\qquad$ .
$=$ 11. $\left(\frac{\pi}{6}, \frac{\pi}{2}\right),\left(\frac{5 \pi}{6}, \frac{3 \pi}{2}\right)$.
Differentiating the given function, we get
$$
y^{\prime}=-2 \sin 2 x+2 \cos x=-2 \cos x(2 \sin x-1) \text {. }
$$

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

As shown in Figure 5.
Combining with the unit circle, we know that the function
$$
y=\cos 2 x+2 \sin x
$$

is monotonically decreasing in the intervals $\left(\frac{\pi}{6}, \frac{\pi}{2}\right),\left(\frac{5 \pi}{6}, \frac{3 \pi}{2}\right)$.