Question

\[\int \cot^n {cosec}^2 \text{ x dx } , n \neq - 1\]

Answer 1

∫ cotⁿ x cosec² x dx
Let cot x = t
⇒ –cosec² x dx = dt
⇒ cosec² x dx = –dt

\[Now, \int \cot^n \text{ x } {cosec}^2  \text  { x dx  }\]
\[ = - \int t^n dt \]
\[ = \frac{- t^{n + 1}}{n + 1} + C\]
\[ = - \frac{\cot^{n + 1} x}{n + 1} + C\]