Undergrad Measurement Values for z-component of Angular Momentum

[MoAli]

Given a wave function $$\Psi(r,\theta,\phi)=f(r)\sin^2(\theta)(2\cos^2(\phi)-1-2i*\sin(\phi)\cos(\phi))$$ we are trying to find what a measurement of angular momentum of a particle in such wave function would yield.
Attempts were made using the integral formula for the Expectation Value over a spherical volume: $$<\hat L_z> = \iiint\Psi^*(\hat L_z)\Psi dV$$ where $$dV=r^2\sin(\theta)drd\theta d\phi$$ and $$\hat L_z=-i\hbar (\frac{\partial}{\partial \phi} ).$$ The Integral seemed really difficult to clear up and get a valid expression, which caused a doubt about whether the approach is incorrect in the first place. Any suggestions on an efficient or smarter way of approaching those types of problems?
Thank you!

 

[blue_leaf77]

It's more useful if you try to think the wavefunction you are given with as a superposition of eigenfunctions of $L_z$. At the same time try to get familiar with some low order spherical harmonics (e.g. in https://en.wikipedia.org/wiki/Table_of_spherical_harmonics) and recognize how they typically depend on the angles $\theta$ and $\phi$ for a given pair of $l$ and $m$.