Example 10. Find the integral $\int \frac{d x}{1+4 x^{2}}$.
Solution.

$$
\int \frac{d x}{1+4 x^{2}}=\int \frac{d x}{1+(2 x)^{2}}=\frac{1}{2} \int \frac{d(2 x)}{1+(2 x)^{2}}=\frac{1}{2} \operatorname{arctg} 2 x+C
$$