Example 5. Find the integral $\int \frac{d x}{\sqrt{5-4 x-x^{2}}}$.
Solution. Completing the square trinomial to a perfect square and using formula (4.23), we find

$$
\begin{aligned}
\int \frac{d x}{\sqrt{5-4 x-x^{2}}} & =\int \frac{d x}{\sqrt{-\left(x^{2}+4 x+4-4-5\right)}}=\int \frac{d x}{\sqrt{-(x+2)^{2}+9}}= \\
& =\int \frac{d(x+2)}{\sqrt{3^{2}-(x+2)^{2}}}=\arcsin \frac{x+2}{3}+C
\end{aligned}
$$