Integral with sine, cosine, and rational function

[Ravendark]

Homework Statement

I would like to compute the following integral:
$$I = \int\limits_0^\pi \mathrm{d}\theta \, \frac{\sin^2 \theta}{a^2 + b^2 - 2 \sqrt{ab} \cos \theta}$$
where $a,b \in \mathbb{R}_+$.

2. The attempt at a solution
Substitution $x = \cos \theta$ yields
$$I = \int\limits_{-1}^1 \mathrm{d}x \, \frac{\sqrt{1 - x^2}}{a^2 + b^2 - 2 \sqrt{ab} \, x} \; .$$
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:
$$I = \int\limits_0^\pi \mathrm{d}\theta \, \frac{\sin^2 \theta}{a^2 + b^2 - 2 \sqrt{ab} \cos \theta}$$
where $a,b \in \mathbb{R}_+$.

2. The attempt at a solution
Substitution $x = \cos \theta$ yields
$$I = \int\limits_{-1}^1 \mathrm{d}x \, \frac{\sqrt{1 - x^2}}{a^2 + b^2 - 2 \sqrt{ab} \, x} \; .$$
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