Question

$\int {\cot }^n{\mathrm{cosec}}^{2}xdx,n\ne -1$

Answer 1

∫ cotⁿ x cosec² x dx
Let cot x = t
⇒ –cosec² x dx = dt
⇒ cosec² x dx = –dt
$$\begin{gathered} \mathrm{Now},\int {\cot }^{n}x{\mathrm{cosec}}^{2}xdx \\ =-\int t^{n}dt \\ =\frac{-t^{n+1}}{n+1}+C \\ =-\frac{{\cot }^{n+1}x}{n+1}+C \end{gathered}$$