The integral inversion curve (IIC) is the contour of all thermodynamic states (P,T) which exhibit a zero integral cooling under Joule-Thomson (isenthalpic) expansion from a pressure, P, down to ambient atmospheric pressure. This is an alternative and complementary presentation of the Joule-Thomson inversion phenomena. The traditional inversion curve is a differential one, namely, it distinguishes between states for which infinitesimal isenthalpic expansion results in heating and in cooling. The IIC has a peaking pressure in the (P,T) plane. This is the highest pressure for which integral cooling is still possible. However, most of the gases solidify before the peaking pressure is reached, or the peaking pressure is too high and not covered by the measurements collected so far. The quantum gases, helium-3, helium-4, hydrogen and deuterium comprise the exceptional group which enables the study and verification of the predicted peaking pattern of the IIC. The helium-4 and hydrogen IlCs are obtained through an available numerical code of thermophysical properties. In addition, the peaking condition in terms of the heat capacities is verified. The IlCs of helium-3 and deuterium are evaluated through advanced equations of state based on 16 parameters and 24 parameters. By analogy to the traditional maximum inversion curve, the maximum integral inversion reduced pressures are determined to be about 40 for helium-4 and hydrogen, 32 for deuterium and 28 for helium-3. Helium-4 exhibits the widest reduced domain of integral cooling states.