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