Answer
$$0$$
Work Step by Step
$$\eqalign{
& \int_{ - 10}^{10} {\frac{x}{{\sqrt {200 - {x^2}} }}dx} \cr
& {\text{testing for symmetry}} \cr
& f\left( x \right) = \frac{x}{{\sqrt {200 - {x^2}} }} \cr
& f\left( { - x} \right) = \frac{{\left( { - x} \right)}}{{\sqrt {200 - {{\left( { - x} \right)}^2}} }} \cr
& f\left( { - x} \right) = - \frac{x}{{\sqrt {200 - {x^2}} }} \cr
& f\left( { - x} \right) = - f\left( x \right){\text{ so the function }}\frac{x}{{\sqrt {200 - {x^2}} }}{\text{ is odd}} \cr
& {\text{Use the theorem 5}}{\text{.4}} \cr
& {\text{If }}f\left( x \right){\text{ is odd}}{\text{, }}\int_{ - a}^a {f\left( x \right)dx} = 0 \cr
& then \cr
& \int_{ - 10}^{10} {\frac{x}{{\sqrt {200 - {x^2}} }}dx} = 0 \cr} $$
