I feel quite dumb for asking this, but it's getting a bit frustrating.
I have a function which depends on $\theta$ and $\phi$ in spherical coordinates:
$ f(\theta, \phi) = \sin^2 \theta \mathcal{L} + \left( \cos^2\theta +\sin^2\phi \sin^2\theta \right) \mathcal{M} $
where $ \mathcal{L}$ and $ \mathcal{M}$ are positive constants.
I want to average this quantity over uniformly random distributed directions. To do this I think of $\theta$ and $\phi$ as two uniform distributed random variables and evaluate
$f^\text{rand} = \int _{-\infty}^{\infty} \, d\phi \int _{-\infty}^{\infty} d \theta\, U(\phi) U(\theta) f(\theta,\phi) = \int _{0}^{2\pi} \, \frac{1}{2\pi} d\phi \,\int _{0} ^{\pi} \frac{1}{\pi} d \theta \, f(\theta,\phi) $
Is this correct? If not, what am I missing?