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-   -   How can I integrate cos(x)^n where n is real? (https://www.askmehelpdesk.com/showthread.php?t=460075)

  • Mar 24, 2010, 10:40 AM
    fairandsquare
    How can I integrate cos(x)^n where n is real?
    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
  • Mar 24, 2010, 11:37 AM
    Unknown008

    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?
  • Mar 24, 2010, 12:43 PM
    fairandsquare
    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
  • Mar 25, 2010, 08:52 AM
    galactus

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



    Is this what you mean?
  • Mar 25, 2010, 05:35 PM
    fairandsquare

    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
  • Mar 26, 2010, 03:29 PM
    galactus

    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:

  • Mar 29, 2010, 05:45 PM
    fairandsquare

    Hi,

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

    Cheers

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