Integration of odd power of cotangent multiplied by odd power of cosecant

[gzAbc123]

Homework Statement

Describe the strategy you would use to (integrate:
cot^m x)(csc^n x)dx, if m and n are odd.

Homework Equations

I know the integral of cosecant is ln |sec x + tan x| + C

I also know the integral of cotangent is ln |sinx| + C

But I have no clue how this would apply to odd powers and multiplying them together.

The Attempt at a Solution

I know how to multiply odd powers of sine and cosine, but for cosecant and cotangent, I have no clue where to get started. The question isn't asking me to actually integrate, but just to describe how I would integrate. Does this integration parallel the corresponding rules for odd powers and multiplication of tanx and secx? Help, please.

 

[lubuntu]

Doesn't your book describe how to do this? thing of an identity that relates the two trig functions, also your not worried about the integral of them individually in this case you want to think about their deviates so you can use substitution...

 

[gzAbc123]

Hey,
No my book only mentions how to solve this for two even-powered; and one odd and one even. It doesn't even give any hints about how to integrate when there are two odd-powered cosecant and cotangent functions being multiplied.

Do you have any other suggestions?

 

[carlodelmundo]

Hi gzAbc123:

Check this link out:

http://books.google.com/books?id=Cl...X&oi=book_result&resnum=5&ct=result#PPA322,M1

Look especially for the "Steven and Todd Rule" and the Odd Man Out Rule... On page 343 Example #2... this is exactly like your problem.

This will help you with these types of integration.

 

[gzAbc123]

Thanks for the link :).

The only problem is not the Steven and Todd rule don't seem to apply for cosecant AND cotangent used in the same equation. Is it there somewhere?

 

[carlodelmundo]

Thanks for the link :).

The only problem is not the Steven and Todd rule don't seem to apply for cosecant AND cotangent used in the same equation. Is it there somewhere?

Did you see page 343 Problem #2?

 

[Hurkyl]

Does this integration parallel the corresponding rules for odd powers and multiplication of tanx and secx?

More than just parallel; armed with the power of trig identities, you can make them the same problem.

 

[gzAbc123]

It says pages 343-344 are not part of this book review... what the?

 

[carlodelmundo]

My bad. Pp. 323 #2

 

[gzAbc123]

But isn't that question for tangent and secant?

Is it basically the same set of steps for cotangent or cosecant? Or is there a few steps that would needed to be added?

 

[carlodelmundo]

But isn't that question for tangent and secant?

Is it basically the same set of steps for cotangent or cosecant? Or is there a few steps that would needed to be added?

Honestly... it's the same steps. The only differences b/w cot and csc vs. tan and cot is that... the derivatives/anti derivatives must take into account the negative (-).