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The acoustic, optic, and surface polar optic phonons are the three important intrinsic and extrinsic phononic modes that increasingly populate graphene on a substrate with rising temperatures; the coupling of which with photoexcited hot carriers in the equipartition regime provides significant pathways for electron-phonon relaxation. In this paper, we theoretically investigate the relative significance of the three phononic modes in electron scattering and cooling phenomena in single layer graphene, including their comparison with supercollision driven power loss, and obtain analytical formulae on the energy dependence of electron-phonon scattering rate and cooling power in the Boltzmann transport formalism. The obtained analytical solutions not only closely reproduce the results for scattering rate and cooling power, as that obtained from the earlier reported numerically tractable integral forms, but also enable us to derive closed-form formulae of the cooling time and thermal conductance. The important role of Pauli blocking that prevents transition to filled energy states has also been elucidated in the estimation of the scattering rate and cooling power density for all the three modes. The obtained formulae provide a better insight into the dynamics of hot electron phenomena giving an explicit view on the interplay of the different variables that affect the transport quantities under investigation. The formulae can also be potentially useful for performance optimization of transport quantities in numerical optimization methods since the first and second-order derivatives are easily deducible from these formulae.