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