Answer
$arcsin(\frac{x+2}{2})+C$
Work Step by Step
$\int\frac{dx}{\sqrt (-x^{2}-4x)}$
To solve this integral, we must manipulate the denominator by completing the square:
$-x^{2}-4x$
$-1(x^{2}+4x)$
$-1(x^{2}+4x+4-4)$
$-x^{2}-4x-4+4$
$4-x^{2}-4x-4$
$4-(x^{2}+4x+4)$
$4-(x+2)^{2}$
$\int\frac{dx}{\sqrt (4-(x+2)^{2})}$
Now with the manipulated denominator, we can identify that using $arcsin$ would solve this integral. The formula for $arcsin$ is:
$\int\frac{du}{\sqrt (a^{2}-u^{2})}=arcsin(\frac{u}{a})+C$
$u=x+2$
$du=dx$
$a=2$
$arcsin(\frac{x+2}{2})+C$
