Answer
\[ = \ln \left| {2 + \sqrt 3 } \right|\]
Work Step by Step
\[\begin{gathered}
\int_{\frac{\pi }{6}}^{\frac{\pi }{2}} {\frac{{dy}}{{\sin y}}} \hfill \\
\hfill \\
use\,\,\,\frac{1}{{\sin y}} = \csc y \hfill \\
\hfill \\
then \hfill \\
\int_{\frac{\pi }{6}}^{\frac{\pi }{2}} {\frac{{dy}}{{\sin y}}} = \int_{\frac{\pi }{6}}^{\frac{\pi }{2}} {\csc ydy} \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= \left. { - \ln \left| {\csc \,y + \cot \,y} \right|} \right|_{\frac{\pi }{6}}^{\frac{\pi }{2}} \hfill \\
\hfill \\
evaluate\,\,the\,\,limits \hfill \\
\hfill \\
= \ln \left| {\csc \frac{\pi }{6} + \cot \frac{\pi }{6}} \right| - \ln \left| {\csc \frac{\pi }{2} + \cot \frac{\pi }{2}} \right| \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \ln \left| {2 + \sqrt 3 } \right| - \ln \left| {1 + 0} \right| \hfill \\
\hfill \\
= \ln \left| {2 + \sqrt 3 } \right| \hfill \\
\hfill \\
\end{gathered} \]
