Question

This is probably the wrong section but:

How does one find the integral of csc^10(x)cot(x)dx?

Answer 1

csc x = (sin x)^-1
\[ \frac{d}{dx} \csc x = \frac{d}{dx} (\sin x)^{-1}=-( \sin x)^{-2} \cos x = - \frac{1}{\sin x} \frac{\cos x }{\sin x}= - \csc x \cot x\]

let u = csc x so that \(du = - \csc x \cot x\ dx\)
and
\[ \int \csc^{10}(x) \cot(x)\ dx =- \int \csc^{9}(x) (- \csc x\cot(x))\ dx \\
=- \int u^9 du
\]

Answer 2

Thank you!