How to Integrate Cotangent and Cosecant with Odd Powers?

[adelaide87]

Describe the method you would use to integrate

cot^a(x) csc^b(x) dx

If a and b are odd?

An explanation of the strategy would be a huge help!

 

[Whitishcube]

I would definitely try to put everything into terms of sin and cos.

cot^a(x) = Cos^a(x) / sin^a(x)

csc^b(x) = 1/sin^b(x)

this is where i would start. from there, you can sum the powers of the sin's in the denominators and try u substitution or something.

 

[Mark44]

Describe the method you would use to integrate

cot^a(x) csc^b(x) dx

If a and b are odd?

An explanation of the strategy would be a huge help!

Take out one factor each of csc x and cot x, leaving you with
$$\int cot^{a-1}(x)csc^{b - 1}(x)~csc(x)cot(x)dx$$

Use the identity cot²(x) + 1 = csc²(x) (or equivalently, cot²(x) = csc²(x) - 1) to replace the cot^{a - 1} factor.

At that point you'll have a sum of terms that involve various powers of csc²(x) and you can use an ordinary substitution, with u = csc(x).