Answer
\[ = \frac{\theta }{2} - \frac{1}{4}\sin \,\left( {2\theta + \frac{\pi }{3}} \right) + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {{{\sin }^2}\,\left( {\theta + \frac{\pi }{6}} \right)} \,d\theta \hfill \\
\hfill \\
use\,\,the\,identity\,\,{\sin ^2}x = \frac{{1 - \cos 2x}}{2} \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
= \int_{}^{} {\frac{{1 - \cos \,\left( {2\theta + \frac{\pi }{3}} \right)}}{2}} \,d\theta \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= \int_{}^{} {\frac{1}{2}\,d\theta } - \frac{1}{4}\int_{}^{} {\cos \,\left( {2\theta + \frac{\pi }{3}} \right)\,\left( 2 \right)\,d\theta } \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \frac{\theta }{2} - \frac{1}{4}\sin \,\left( {2\theta + \frac{\pi }{3}} \right) + C \hfill \\
\end{gathered} \]
