Answer
$$\ln 27$$
Work Step by Step
$$\eqalign{
& \int_{\pi /2}^\pi {2\cot \frac{\theta }{3}} d\theta \cr
& = 2\int_{\pi /2}^\pi {\cot \frac{\theta }{3}} d\theta \cr
& {\text{Integrate with the formula }}\cr
& \int {\cot ax} dx = \frac{1}{a}\ln \left| {\sin ax} \right| + C \cr
& = \frac{2}{{1/3}}\left( {\ln \left| {\sin \frac{\theta }{3}} \right|} \right)_{\pi /2}^\pi \cr
& {\text{Use the fundamental theorem of calculus }}\int_a^b {f\left( x \right)} dx = F\left( b \right) - F\left( a \right).\,\,\,\,\left( {{\text{see page 281}}} \right) \cr
& = 6\left( {\ln \left| {\sin \frac{\pi }{3}} \right| - \ln \left| {\sin \frac{\pi }{6}} \right|} \right) \cr
& {\text{Simplifying, we get:}} \cr
& = 6\left( {\ln \left( {\frac{{\sqrt 3 }}{2}} \right) - \ln \left( {\frac{1}{2}} \right)} \right) \cr
& = 6\left( {\ln \sqrt 3 } \right) \cr
& = \ln {\left( {\sqrt 3 } \right)^6} \cr
& = \ln 27 \cr} $$
