Evaluate the definite integral _ 6 ^ 4 ecxdx - Vidyadip

Question

Evaluate the definite integral $\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{4}} {\cos ecxdx} $

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Answer

$\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{4}} {\cos ecxdx} $

$= \left( {\log \left| {\cos ecx - \cot x} \right|} \right)_{\frac{\pi }{6}}^{\frac{\pi }{4}}$

$= \log \left| {\cos ec\frac{\pi }{4} - \cot \frac{\pi }{4}} \right| - \log \left| {\cos ec\frac{\pi }{6} - \cot \frac{\pi }{6}} \right|$

$= \log \left| {\sqrt 2 - 1} \right| - \log \left| {2 - \sqrt 3 } \right|$

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

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

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