Answer
$-\frac{3\sqrt 3}{2\pi} .$
Work Step by Step
We have
\begin{align*}
\int_{0}^{1} \cos \left(\frac{\pi}{3}(t+2)\right) d t&=\frac{3}{\pi}\int_{0}^{1} \frac{\pi}{3}\cos \left(\frac{\pi}{3}(t+2)\right) d t\\
&=\frac{3}{\pi} \sin\left(\frac{\pi}{3}(t+2)\right)|_{0}^{1}\\
&=\frac{3}{\pi} \left( \sin\left(\pi\right) - \sin \frac{2\pi}{3}\right)\\
&= -\frac{3\sqrt 3}{2\pi} .
\end{align*}
