Mathematics

Evaluate the following integral:
$$\displaystyle \int { co\sec { x } \log { \left( co\sec { x } -\cot { x }  \right)  }  } dx\quad $$


SOLUTION
We've,
$$\dfrac{d}{dx}\{\log (cosec x-\cot x\}=\dfrac{-cosec x.\cot x+cosec^2 x}{cosec x-\cot x}=cosec x.$$.....(1).

Now,

$$\int { cosec { x } .\log { \left( cosec { x } -\cot { x }  \right)  }  } dx$$

$$=\int { \log { \left( cosec { x } -\cot { x }  \right)  }  } d\{\log(cosec x-\cot x)\}$$ [ Using (1)]

$$=\dfrac{\{\log(cosec x-\cot x)\}^2}{2}+c$$. [ Where $$c$$ is integrating constant]
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Subjective Medium Published on 17th 09, 2020
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