Answer
$$ - 4\sqrt 2 + 8$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {4\sin \left( {x/2} \right)} dx \cr
& u = \frac{x}{2},{\text{ then }}du = \frac{1}{2}dx,{\text{ }}2du = dx \cr
& {\text{With this substitution}}{\text{,}} \cr
& {\text{if }}x = \pi /2,{\text{ }}u = \frac{{\pi /2}}{2} = \frac{\pi }{4} \cr
& {\text{if }}x = 0,{\text{ }}u = \frac{0}{2} = 0 \cr
& {\text{so}} \cr
& \int_0^{\pi /2} {4\sin \left( {x/2} \right)} dx = \int_0^{\pi /4} {4\sin \left( {x/2} \right)} \left( {2du} \right) \cr
& = 8\int_0^{\pi /4} {\sin udu} \cr
& {\text{find the antiderivative}} \cr
& = 8\left( { - \cos u} \right)_0^{\pi /4} \cr
& {\text{part 1 of fundamental theorem of calculus}} \cr
& = 8\left( { - \cos \left( {\frac{\pi }{4}} \right) + \cos \left( 0 \right)} \right) \cr
& = 8\left( { - \frac{{\sqrt 2 }}{2} + 1} \right) \cr
& = - 4\sqrt 2 + 8 \cr} $$
