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Integral help

  1. May 4, 2012 #1

    I need some help to calculate this integral:
    [tex]\int _0^{2\pi}\frac{x^n}{\sqrt{1-m\cos x}}dx[/tex], where 0<m<1.

    What I've already tried:
    took the binomial series of (1-m cos(x))^(-1/2), this results in integrals like

    [tex]\int_0^{2\pi} x^n(\cos x)^k dx[/tex]

    After this I've replaced cos(x)^k as a polynomial of cos(r*x) (r=1,2,...,k). With this I've managed to get a formula (involving two summas), but it is so ugly that I cannot use them in any furhter calculations.

    (sorry, I don't know how to make formulas in PF, so I've inserted the LaTex code of it)

    Thank You!
    Last edited by a moderator: May 4, 2012
  2. jcsd
  3. May 4, 2012 #2


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    You integrate
    [tex]\int x^ncos^k(x) dx[/tex]
    using integration by parts- n times.

    (I have replaced your "$" with [ tex ] to start and [ /tex ] to end the LaTeX- without the spaces.)
  4. May 4, 2012 #3


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    Also if you are curious you can use the fact that:

    [itex]cos(x) = \frac{e^{ix} + e^{-ix}}{2}[/itex] and you can take that to whatever power you want. This even works for non-integral powers where the result is valid.
  5. May 4, 2012 #4
    Thank you for your help with the formula. However, I don't see how integration by parts works in this case, because while differentiating the cosine term, I will have some ugly terms.

    I think that this is exactly the same as what I've done (at least for integer k-s)
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