Answer
$ - \frac{3}{\pi }\cos \left( {\frac{{\pi t}}{3}} \right) + C$
Work Step by Step
$$\eqalign{
& \int {\sin \left( {\frac{{\pi t}}{3}} \right)} dt \cr
& {\text{Let }}u = \frac{{\pi t}}{3},{\text{ then }}du = \frac{\pi }{3}dt,{\text{ }}\frac{3}{\pi }du = dt \cr
& {\text{Applying the substitution}}{\text{, we obtain}} \cr
& \int {\sin \left( {\frac{{\pi t}}{3}} \right)} dt = \int {\sin u} \left( {\frac{3}{\pi }} \right)du \cr
& = \frac{3}{\pi }\int {\sin u} du \cr
& {\text{Integrating}} \cr
& = \frac{3}{\pi }\left( { - \cos u} \right) + C \cr
& {\text{Write in terms of }}t,{\text{ substitute }}u = \frac{{\pi t}}{3} \cr
& = \frac{3}{\pi }\left( { - \cos \left( {\frac{{\pi t}}{3}} \right)} \right) + C \cr
& = - \frac{3}{\pi }\cos \left( {\frac{{\pi t}}{3}} \right) + C \cr} $$
