Answer
$${\sin ^{ - 1}}\left( {\frac{x}{2}} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{dx}}{{\sqrt {4 - {x^2}} }}} \cr
& {\text{substitute }}x = 2\sin \theta ,{\text{ }}dx = 2\cos \theta d\theta \cr
& \int {\frac{{dx}}{{\sqrt {4 - {x^2}} }}} = \int {\frac{{2\cos \theta d\theta }}{{\sqrt {4 - {{\left( {2\sin \theta } \right)}^2}} }}} \cr
& = \int {\frac{{2\cos \theta d\theta }}{{\sqrt {4\left( {1 - {{\sin }^2}\theta } \right)} }}} \cr
& = \int {\frac{{2\cos \theta d\theta }}{{2\sqrt {{{\cos }^2}\theta } }}} \cr
& = \int {d\theta } \cr
& {\text{find antiderivative}} \cr
& = \theta + C \cr
& x = 2\sin \theta \to \theta = {\sin ^{ - 1}}\left( {\frac{x}{2}} \right) \cr
& = {\sin ^{ - 1}}\left( {\frac{x}{2}} \right) + C \cr} $$
