Question

\[\text{ ∫  cosec x  log}      \left( \text{cosec x} - \cot x \right) dx\]

Answer 1

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