Indeed the enthalpy H=U+PV=integral[(Cp)dT] where H is the enthalpy, U is the internal energy, P is the pressure, V the volume, Cp the specific heat capacity at constant pressure and T the temperature.
The internal energy can be calculated from a formula using statistical mechanics, however this can be quite involved. For simple systems such as an ideal gas or a spin system the solution is straightforward but it can get rather complicated.
The calorific value of a substance is essentially the energy that is released upon burning the substance. This ties in with the enthalpy of combustion.