Answer
$$2\sin \left( {\frac{x}{2} + \frac{\pi }{3}} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\cos \left( {\frac{x}{2} + \frac{\pi }{3}} \right)} dx \cr
& {\text{substitute }}u = \frac{x}{2} + \frac{\pi }{3},{\text{ }}du = \frac{1}{2}dx \cr
& \int {\cos \left( {\frac{x}{2} + \frac{\pi }{3}} \right)} dx = \int {\cos u} \left( {2du} \right) \cr
& = 2\int {\cos u} du \cr
& {\text{find the antiderivative}} \cr
& = 2\sin u + C \cr
& {\text{ replacing }}u = \frac{x}{2} + \frac{\pi }{3} \cr
& = 2\sin \left( {\frac{x}{2} + \frac{\pi }{3}} \right) + C \cr} $$
