Answer
$$ - \ln \left( {\frac{{\sqrt 2 }}{2}} \right)$$
Work Step by Step
$$\eqalign{
& \int_{\pi /4}^{\pi /2} {\cot x} dx \cr
& {\text{integrate by using the basic integration rule }}\int {\cot x} dx = \ln \left| {\sin x} \right| + C\,\,\,\left( {{\text{page 694}}} \right) \cr
& then \cr
& = \left( {\ln \left| {\sin x} \right|} \right)_{\pi /4}^{\pi /2} \cr
& {\text{use fundamental theorem of calculus }}\int_a^b {f\left( x \right)} dx = F\left( b \right) - F\left( a \right).\,\,\,\,\left( {{\text{see page 388}}} \right) \cr
& = \ln \left| {\sin \frac{\pi }{2}} \right| - \ln \left| {\sin \frac{\pi }{4}} \right| \cr
& {\text{simplifying}} \cr
& = \ln \left( 1 \right) - \ln \left( {\frac{{\sqrt 2 }}{2}} \right) \cr
& = - \ln \left( {\frac{{\sqrt 2 }}{2}} \right) \cr} $$
