Separation of Variables for Laplace's Equation in Spherical Coordinates

When one solves Laplace’s equation for charge distributed on a spherical surface: $nabla^2V = 0 $ in spherical coordinates, one can obtain a solution that is a sum of Legendre Polynomials, thus having a longitude angular $theta$ dependence. However, solving the potential using the equation $V = frac{1}{4piepsilon_0}int frac{sigma}{r}da$ seems to negate the $theta$ dependence, since we integrate $theta$ from 0 to $pi$. I’m not sure how to bring about the angular dependence of the potential in such problems. Thank you for any and all help.