Question

Help integrate: cot^3(x)

Answer 1

`\int {cot^(3) x} \, dx = \int { (cos^(3) )/(sin^(3) x) } \, dx = \\ = \int {(1-sin ^(2)) cosx}/sin^(3) x \, dx = \\ \int { (cosx)/(sin^(3) x) } \, dx - \int { (cos x)/(sin x) } \, dx`
We will use u-substitution:
u = sin x, du = cos x dx
=
`\int { (du)/(u^(3) )} - \int { (du)/(u) }=- (1)/(2u^(2) ) - ln (u) = \\ =- (1)/(2 sin^(2) x) +ln (sin x)+C`