Example 14. Find $\int(1+2 \sin x)^{2} d x$.
Solution. After squaring, we obtain the non-tabular integral $\int \sin ^{2} x d x$, which can be reduced to tabular ones using the power reduction formula: $\sin ^{2} x=\frac{1}{2}(1-\cos 2 x)$. Thus, we have:

$$
\begin{aligned}
& \int(1+2 \sin x)^{2} d x=\int(1+4 \sin x+2(1-\cos 2 x)) d x= \\
& \quad=3 \int d x+4 \int \sin x d x-2 \int \cos 2 x d x=3 x-4 \cos x-\sin 2 x+C
\end{aligned}
$$