Question

Trigonometric Integrals with cosecant and cotangent?

`int csc^3(x)cot(x)`

Answer 1

`int \ csc^3x \ cotx \ dx = -1/3csc^3x + C`

Explanation:

We seek the integral:

`I = int \ csc^3x \ cotx \ dx`

First we replace all trig terms with sine and cosine; so:

`I = int \ 1/sin^3x \ cosx/sinx \ dx`
`\ \ = int \ cosx/sin^4x \ dx`

Now, we can perform a substitution. Let:

`u = sinx => (du)/dx = cosx`

And so we can substitute into the integral to get:

`I = int \ 1/u^4 \ du`

Which is a standard integral, and so:

`I = -1/(3u^3) + C`

And restoring the substitution we get:

`I = -1/(3sin^3x) + C`
`\ \ = -1/3csc^3x + C`