Coarse-graining (CG) procedures provide computationally efficient methods for investigating the corresponding long time- and length-scale processes. In the bottom-up approaches, the effective interactions between the CG sites are obtained using the information from the atomistic simulations, but reliable CG procedures are required to preserve the structure and thermodynamics. In this regard, the integral equation coarse-graining (IECG) method is a promising approach that uses the first-principles Ornstein-Zernike equation in liquid state theory to determine the effective potential between CG sites. In this work, we present the details of the IECG method while treating the density as an intrinsic property and active variable of the CG system. Performing extensive simulations of polymer melts, we show that the IECG theory/simulation and atomistic simulation results are consistent in structural properties such as the pair-correlation functions and form factors, and also thermodynamic properties such as pressure. The atomistic simulations of the liquids show that the structure is largely sensitive to the repulsive part of the potential. Similarly, the IECG simulations of polymeric liquids show that the structure can be determined by the relatively short-range CG repulsive interactions, but the pressure is only accurately determined once the long-range, weak CG attractive interactions are included. This is in agreement with the seminal work by Widom on the influence of the potential on the phase diagram of the liquid [Widom, B. Science 1967, 157, 375-382]. Other aspects of the IECG theory/simulations are also discussed.