Answer
$0$
Work Step by Step
$$\eqalign{
& \int_{ - \pi /3}^{\pi /3} {{x^4}\sin x} dx \cr
& {\text{Let }}f\left( x \right) = {x^4}\sin x{\text{ and evaluate }}f\left( { - x} \right) \cr
& f\left( { - x} \right) = {\left( { - x} \right)^4}\sin \left( { - x} \right) \cr
& f\left( { - x} \right) = {x^4}\sin \left( { - x} \right) \cr
& f\left( { - x} \right) = - {x^4}\sin x \cr
& f\left( x \right) = - f\left( x \right),{\text{ therefore }}{x^4}\sin x{\text{ is an odd function}}{\text{, using}} \cr
& {\text{the property }}\int_{ - a}^a {f\left( x \right)} dx = 0,{\text{ if }}f\left( x \right){\text{ is odd}}{\text{, we obtain}} \cr
& \int_{ - \pi /3}^{\pi /3} {{x^4}\sin x} dx = 0 \cr} $$