Solution to a definite integral

Asked Feb 03 '24 at 19:17

Active Feb 03 '24 at 19:26

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I have the following definite integral.

$$\int_{0}^{2\pi} \left(\frac{1}{\left(1-B\cos(x)\right)^2}\right)dx$$

I tried inputting into Wolfram and it exceeded the computation time. Anyone know what or how to get a solution (if there is one).

asked Feb 03 '24 at 19:17

What is $B$ in here? – ADAM Feb 03 '24 at 19:23
B is a constant – rdemyan Feb 03 '24 at 19:35
Wolfram gets an answer quickly: $2\pi/(1-B^2)^{3/2},$ provided (of course) $|B|\lt 1.$ – whuber Feb 03 '24 at 19:45
This could be solved maybe by Residue Theorem – Joan S. Guillamet F. Feb 03 '24 at 19:54
@whuber - Thanks, but I just tried again with Wolfram and standard computation time was exceeded. Did you input the formula in a different way than what I show in my original post? Or is there a way that I tell Wolfram that B < 1, which it is by the way. – rdemyan Feb 03 '24 at 19:55
One trick is to specify strong assumptions that are still likely to produce a general result. I used Integrate[1 / (1 - Cos[x] b)^2, {x, 0, 2 Pi}, Assumptions -> 0 < b < 1/2]. – whuber Feb 03 '24 at 21:46

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