Integral with sine, cosine, and rational function

[Ravendark]

Homework Statement

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 don't know how to proceed. I have in mind to use the residue theorem somehow, but I don't know if this is applicable here. Can someone give me a hint, please?

 

[Ray Vickson]

Homework Statement

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 don't know how to proceed. I have in mind to use the residue theorem somehow, but I don't know if this is applicable here. Can someone give me a hint, please?

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