Mathematics

Evaluate the following definite integrals:
$$\displaystyle \int _{\pi /6}^{\pi /4} cosec \  x \ dx$$ 


SOLUTION
$$I=\displaystyle \int _{\pi /6}^{\pi /4} cosec \  x \  dx$$

$$=[\log (cosec \  x-\cot x)]_{\pi /6}^{\pi /4} $$

$$=\log \left (cosec \dfrac {\pi}{4} -\cot \dfrac {\pi}{4}\right) -\log \left (cosec \dfrac {\pi}{6}-\cot \dfrac {\pi}{6} \right)$$

$$=\log (\sqrt 2-1)-\log (2-\sqrt 3)$$

$$=\log \left (\dfrac {\sqrt 2-1}{2-\sqrt 3}\right )$$
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Subjective Medium Published on 17th 09, 2020
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