What is the Integral with Symmetries in a Minimization Problem?

[the_phoenix]

Homework Statement

Hi, i need to integrate a function which has a very nice structure. however, in case i cannot do the same, i would need a bound on the value of the integral. The problem is basically a minmization problem with n parameters. Please follow the link to acquaint your self with the problem.
http://img522.imageshack.us/img522/7894/equations640x480.gif

Homework Equations

http://img522.imageshack.us/img522/7894/equations640x480.gif
here, d_i and theta_i are the distances and angles of point (x,y) from (xi,yi)

The Attempt at a Solution

My attempt is as follows:
First of al, i could not directly integrate the function, so i tried to find a bound on it. I used the mod of the function (extension of triangle inequality) on the denominator to eliminate the sine squared in the denominator.
Upon doing that i tried to split the denominator through chebyshev sum inequality but didnt get anywhere.

I hope you can give me some insights into either solving the integral or getting a bound on its value.

thanks and regards

 

[benorin]

Would you elaborate on the geometry of the sum, specifically, where does the $$d_{i}^{-2}d_{j}^{-2}\sin^2 (\theta_{i}-\theta_{j})$$ term come from?