Answer
$$2\arcsin \left( {\frac{{x - 2}}{2}} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{2}{{\sqrt { - {x^2} + 4x} }}} dx \cr
& {\text{Completing the square}} \cr
& \int {\frac{2}{{\sqrt { - {x^2} + 4x} }}} dx = \int {\frac{2}{{\sqrt { - \left( {{x^2} - 4x + 4 - 4} \right)} }}} dx \cr
& = \int {\frac{2}{{\sqrt {4 - {{\left( {x - 2} \right)}^2}} }}} dx \cr
& {\text{Let }}u = x - 2,{\text{ }}x = u + 2,{\text{ }}dx = du,{\text{ }} \cr
& \int {\frac{2}{{\sqrt {4 - {{\left( {x - 2} \right)}^2}} }}} dx = \int {\frac{2}{{\sqrt {4 - {u^2}} }}} du \cr
& {\text{Integrate using basic integration rules}} \cr
& \int {\frac{2}{{\sqrt {4 - {u^2}} }}} du = 2\arcsin \left( {\frac{u}{2}} \right) + C \cr
& {\text{Write in terms of }}x \cr
& = 2\arcsin \left( {\frac{{x - 2}}{2}} \right) + C \cr} $$
