Answer
$$0$$
Work Step by Step
$$\eqalign{
& \int_{ - 2}^2 {{x^9}dx} \cr
& {\text{testing for symmetry}} \cr
& f\left( x \right) = {x^9} \cr
& f\left( { - x} \right) = {\left( { - x} \right)^9} \cr
& f\left( { - x} \right) = - {x^9} \cr
& f\left( { - x} \right) = - f\left( x \right){\text{ so the function }}{x^9}{\text{ is odd}} \cr
& {\text{Use the theorem 5}}{\text{.4}} \cr
& {\text{If }}f\left( x \right){\text{ is odd}}{\text{, }}\int_{ - a}^a {f\left( x \right)dx} = 0 \cr
& then \cr
& \int_{ - 2}^2 {{x^9}dx} = 0 \cr} $$
