# Evaluating a double integral containing absolute value squared of sum of two functions.

I need to evaluate integrals of this form

$\int\limits_{\phi = 0}^{2\pi }\ \int\limits_{\theta = 0}^{\pi} \left | f_1(\theta, \phi)e^{ix} + f_2(\theta, \phi)e^{ix'} \right |^2\sin{\theta} \ d\theta d\phi$

Where,

$f_1$ and $f_2$ are harmonic functions of $\theta$ and $\phi$. Mostly, $\cos^2{\frac{\theta}{2}}$ or $\sin^2{\frac{\theta}{2}}$ and in general, $f_1 \neq f_2$. $x$ and $x'$ are some real constants.

Any help regarding how I can evaluate these integrals is highly appreciated.

-

Hint: If $f_1$ and $f_2$ are real, then $$|f_1 e^{i x}+f_2 e^{i x'}|^2 = f_1^2+2f_1 f_2\cos(x-x') + f_2^2.$$