How can I analytically integrate cos(x)^n where n is real, but n > 1? I know the answer when n is an integer, but the method doesn't seems to apply to real numbers.

Cheers

Well, I don't know how to do it, but from wolframalpha, it's possible...

http://www.wolframalpha.com/input/?i=int+cos(x)^n

See if you can make head or tail of it...

Maybe galactus/ebaines can provide the method to solve this?

Thanks for your help! The hypergeometric function is a bit of a mystery to me as well! I was hoping there would be another method that didn't invole that, but if not, could anyone explain why it is required.

Thanks

You can use integration by parts to find the reduction formula:



Is this what you mean?

Hi,
Thanks for your response. I believe the redustion formula only works when n is an integer. However, I don't think this equation works when n is real, as at some point if say n = 3.514, then at some point we will have to integrate cos(x)^1.514, which, if I am correct, the above equation will not solve.

Cheers

I don't believe there is a catchall closed form for all n.

Here is a fun one though involving Wallis' formula:



for and

For n even and greater than or equal to 2:

Hi,

Thanks for the info. Those two integrals looks interesting, thanks for pointing them out.

Cheers