Question

Evaluate $\int {\cot }^{2}xdx$.

Answer 1

$\int {\cot }^{2}xdx$

We know that ${\text{cosec}}^{2}x-{\cot }^{2}x=1$

Therefore, $\Rightarrow {\cot }^{2}x={\text{cosec}}^{2}x-1$

$\Rightarrow \int {\cot }^{2}xdx=\int ({\text{cosec}}^{2}x-1)dx$

$\Rightarrow \int {\text{cosec}}^{2}xdx-\int 1dx$

This can be written as

$\Rightarrow -\int -{\text{cosec}}^{2}xdx-\int 1dx$

This is a direct integral

$\Rightarrow -\left(\cot x\right)-x+C$

$\int {\cot }^{2}xdx=-\cot x-x+C$