Question

Answer 1

$$\begin{gathered} \int {\cot }^{3}x{\mathrm{cosec}}^{2}xdx \\ \mathrm{Let},\cot x=t \\ \Rightarrow -{\mathrm{cosec}}^{2}x=\frac{dt}{dx} \\ \Rightarrow {\mathrm{cosec}}^{2}xdx=-dt \\ \mathrm{Now},\int {\cot }^{3}x{\mathrm{cosec}}^{2}xdx \\ =\int t^3\left(-dt\right) \\ =\frac{-t^4}{4}+\mathrm{C} \\ =\frac{-{\cot }^{4}x}{4}+\mathrm{C} \end{gathered}$$