Question

Help integrate: cot^3(x)

Answer 1

`\displaystyle\int\cot^3x\,\mathrm dx=\int\cot^2x\cot x\,\mathrm dx=\int(\csc^2x-1)\cot x\,\mathrm dx`

`\displaystyle=\int\csc^2x\cot x\,\mathrm dx-\int\cot x\,\mathrm dx`

For the first integral, substitute
`y=\cot x` so that
`\mathrm dy=-\csc^2x\,\mathrm dx`. This leaves you with two standard integrals,


`=\displaystyle-\int y\,\mathrm dy-\int\cot x\,\mathrm dx`

`=-\frac12y^2+\ln|\cot x+\csc x|+C`

`=-\frac12\cot^2x+\ln|\cot x+\csc x|+C`