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,

$=\displaystyle-\int y\,\mathrm dy-\int\cot x\,\mathrm dx$
$=-\dfrac12y^2+\ln|\cot x+\csc x|+C$
$=-\dfrac12\cot^2x+\ln|\cot x+\csc x|+C$