Example 6. Find the integral $\int \frac{d x}{\sqrt{6-4 x-2 x^{2}}}$.
Solution. As in example 5, we get

$$
\begin{gathered}
\int \frac{d x}{\sqrt{6-4 x-2 x^{2}}}=\int \frac{d x}{\sqrt{-2\left(x^{2}+2 x-3\right)}}= \\
=\int \frac{d x}{\sqrt{-2\left[\left(x^{2}+2 x+1\right)-1-3\right]}}=\frac{1}{\sqrt{2}} \int \frac{d x}{\sqrt{-\left[(x+1)^{2}-4\right]}}= \\
=\frac{1}{\sqrt{2}} \int \frac{d(x+1)}{\sqrt{2^{2}-(x+1)^{2}}}=\frac{1}{\sqrt{2}} \arcsin \frac{x+1}{2}+C
\end{gathered}
$$