Solving Integral: \int_0^{2\pi}\frac{x^n}{\sqrt{1-m\cos x}}dx

[csopi]

Hi,

I need some help to calculate this integral:
$$\int _0^{2\pi}\frac{x^n}{\sqrt{1-m\cos x}}dx$$, 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

$$\int_0^{2\pi} x^n(\cos x)^k dx$$

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!

 

[HallsofIvy]

You integrate
$$\int x^ncos^k(x) dx$$
using integration by parts- n times.

(I have replaced your "$" with [ tex ] to start and [ /tex ] to end the LaTeX- without the spaces.)

 

[chiro]

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

$cos(x) = \frac{e^{ix} + e^{-ix}}{2}$ and you can take that to whatever power you want. This even works for non-integral powers where the result is valid.

 

[csopi]

HallsofIvy:
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.

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