Question

` ∫  {cosec x} / {"cosec x "- cot x} ` dx      

Answer 1

` ∫ ( {"cosec "x} / {"cosec x "- cot x} )` dx 
\[ = \int\frac{\text{cosec x}\left( \text{cosec x }+ \cot x \right)}{\left(\text{ cosec x} - \cot x \right) \left( \text{cosec x }+ \cot x \right)}dx\]
\[ = \int\frac{\text{cosec x} \left( \text{cosec x }+ \cot x \right)}{\left( {\text{cosec}}^2 x - \cot^2 x \right)}dx\]
\[ = \int\left( {\text{cosec}}^2 x + \text{cosec x } \cot x \right)dx\]
\[ = - \cot x - \text{cosec x }+ C\]