Question

Help integrate: cot^3(x)

Answer 1

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