$\int\limits_{\pi /6}^{\pi /4} {\cos ec\,x\,} dx$

: $\int\limits_{\pi /6}^{\pi /4} {\cos ec\,x\,} dx = \left[ {\log \left( {\cos ec\,x - \cot x} \right)} \right]_{\pi /6}^{\pi /4}$

$= \log \left( {\cos ec\cfrac{\pi }{4} - \cot \cfrac{\pi }{4}} \right) - \log \left( {\cos ec\cfrac{\pi }{6} - \cot \cfrac{\pi }{6}} \right)$

$= \log \left( {\sqrt 2 - 1} \right) - \log \left( {2 - \sqrt 3 } \right) = \log \left( {\cfrac{{\left( {\sqrt 2 - 1} \right)}}{{2 - \sqrt 3 }}} \right)$