f[θ_,ϕ_]:=2*(Sqrt[(15/(16π))])Cos[θ]Sin[θ]Cos[ϕ]

Laplacian[f[θ,ϕ],{θ,ϕ},{"Standard",{"Sphere",1}}]

Simplify[-2 (Sqrt[(15/π)]) Cos[θ] Cos[ϕ] Sin[θ]+Csc[θ] (-(1/2) (Sqrt[(15/π)]) Cos[θ] Cos[ϕ]+Cos[θ] ((1/2) (Sqrt[(15/π)]) ((Cos[θ])^2) Cos[ϕ]-(1/2) (Sqrt[(15/π)]) Cos[ϕ] ((Sin[θ])^2)))]

(∫_0^(2π))(∫_0^π)(f[θ,ϕ]*3 (Sqrt[(15/π)]) Cos[θ] Cos[ϕ] Sin[θ]*Sin[θ])\[DifferentialD]θ\[DifferentialD]ϕ