Answer
$$0$$
Work Step by Step
$$\eqalign{
& \int_{ - 2}^2 {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx \cr
& {\text{The integrand }}\frac{{2x}}{{\sqrt {4 - {x^2}} }}{\text{ is not continuous to }}x = \pm 2,{\text{ then}} \cr
& {\text{Improper Integral with an Infinite Limit of Integration}} \cr
& \int_{ - 2}^2 {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx = \mathop {\lim }\limits_{a \to - {2^ + }} \int_{ - 2}^0 {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx + \mathop {\lim }\limits_{b \to {2^ - }} \int_0^b {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx \cr
& {\text{*Computing }}\mathop {\lim }\limits_{a \to - {2^ + }} \int_{ - 2}^0 {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx \cr
& {\text{Where that }}\int {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx = - \int {{{\left( {4 - {x^2}} \right)}^{ - 1/2}}\left( { - 2x} \right)dx} \cr
& = - \frac{{{{\left( {4 - {x^2}} \right)}^{ 1/2}}}}{{1/2}} + C = - 2\sqrt {4 - {x^2}} + C,{\text{ then}} \cr
& \mathop {\lim }\limits_{a \to - {2^ + }} \int_{ - 2}^0 {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx = \mathop {\lim }\limits_{a \to - {2^ + }} \left[ { - 2\sqrt {4 - {x^2}} } \right]_a^0 \cr
& = \mathop {\lim }\limits_{a \to - {2^ + }} \left[ { - 2\sqrt {4 - {0^2}} + 2\sqrt {4 - {a^2}} } \right] \cr
& = \mathop {\lim }\limits_{a \to - {2^ + }} \left[ { - 4 + 2\sqrt {4 - {a^2}} } \right] \cr
& = - 4 + 2\sqrt {4 - {{\left( { - 2} \right)}^2}} \cr
& = - 4 \cr
& {\text{*Computing }}\mathop {\lim }\limits_{b \to {2^ - }} \int_0^b {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx \cr
& \mathop {\lim }\limits_{b \to {2^ - }} \int_0^2 {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx = \mathop {\lim }\limits_{b \to {2^ - }} \left[ { - 2\sqrt {4 - {x^2}} } \right]_0^b \cr
& = \mathop {\lim }\limits_{b \to {2^ - }} \left[ { - 2\sqrt {4 - {b^2}} + 2\sqrt {4 - {0^2}} } \right] \cr
& = \mathop {\lim }\limits_{b \to {2^ - }} \left[ { - 2\sqrt {4 - {b^2}} + 4} \right] \cr
& = - 2\sqrt {4 - {{\left( 2 \right)}^2}} + 4 \cr
& = 4 \cr
& {\text{Therefore,}} \cr
& \int_{ - 2}^2 {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx = - 4 + 4 \cr
& \int_{ - 2}^2 {\frac{{2x}}{{\sqrt {4 - {x^2}} }}} dx = 0 \cr} $$
