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Integral with sine, cosine, and rational function

  1. Nov 17, 2015 #1
    1. The problem statement, all variables and given/known data
    I would like to compute the following integral:
    [tex]I = \int\limits_0^\pi \mathrm{d}\theta \, \frac{\sin^2 \theta}{a^2 + b^2 - 2 \sqrt{ab} \cos \theta} [/tex]
    where ##a,b \in \mathbb{R}_+##.

    2. The attempt at a solution
    Substitution ##x = \cos \theta## yields
    [tex]
    I = \int\limits_{-1}^1 \mathrm{d}x \, \frac{\sqrt{1 - x^2}}{a^2 + b^2 - 2 \sqrt{ab} \, x} \; .
    [/tex]
    Now I dont know how to proceed. I have in mind to use the residue theorem somehow, but I dont know if this is applicable here. Can someone give me a hint, please?
     
  2. jcsd
  3. Nov 17, 2015 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Assuming that a,b > 0, Maple 11 gets the integral as

    upload_2015-11-17_1-40-30.png
     
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