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Question
Solution
\[\ ∫ cosec x \cdot \log \left( cosec x - \cot x \right) dx\]
\[\text{Let log }\left( \text{cosec x} - \text{cot x }\right) = t\]
\[ \Rightarrow \frac{\left( \text{- cosec x cot x} + {cosec}^2 x \right)}{\left( \text{cosec}\text{ cosec x - cot x }\right)} = \frac{dt}{dx}\]
`⇒ (("cosec" x - cot x ) / ("cosec x" - cot x)) × "cosec" x dx = dt `
\[ \Rightarrow \text{cosec x dx }= dt\]
\[Now, \text{ ∫ cosec x} \cdot \log \left( \text{cosec x }- \cot x \right) dx\]
\[ = \ ∫ t . dt\]
\[ = \frac{t^2}{2} + C\]
\[ = \frac{\left\{ \text{log} \left| \text{cosec x }- \text{cot x} \right| \right\}^2}{2} + C\]
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