Example 12. Find $\int \frac{\sqrt{4-x^{2}}}{x^{2}} d x$.
Solution. Let $x=2 \sin t, d x=2 \cos t d t, \sqrt{4-x^{2}}=2 \cos t$. We arrive at the integral

$$
\begin{aligned}
\int \frac{\sqrt{4-x^{2}}}{x^{2}} d x=\int \frac{\cos ^{2} t}{\sin ^{2} t} d t=-\operatorname{ctg} t- & t+C= \\
& =-\operatorname{ctg}\left(\arcsin \frac{x}{2}\right)-\arcsin \frac{x}{2}+C
\end{aligned}
$$