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Line intensity mapping (LIM) experiments probing the nearby universe can expect a considerable amount of cosmic infrared background (CIB) contiuum emission coming from near and far-infrared galaxies. For the purpose of using the LIM data to constrain the star formation rate (SFR), we argue that the CIB continuum - traditionally treated as contamination - can be combined with the LIM signal to enhance the SFR constraints achievable. We first present a power spectrum model that is capable of joining continuum and line emissions that assume the same prior SFR model. We subsequently analyze the effectiveness of the joint model in the context of the EXperiment for Cryogenic Large-Aperture Intensity Mapping (EXCLAIM), which utilizes the [CII] molecular line to study the SFR. We numerically compute the theoretical power spectra according to our model and the EXCLAIM survey specifics, and perform Fisher analysis to obtain SFR parameter constraints. We find that although the joint model has no considerable advantage over LIM alone assuming the current survey level of EXCLAIM, its effects become significant when we consider more optimistic values of survey resolution and angular span that are expected of future LIM experiments. By manipulating the Fisher formalism, we show that the CIB is not only an additional SFR sensitive signal, but also serves to break the SFR parameter degeneracy that naturally emerges from the [CII] Fisher matrix. For this reason, addition of the CIB will allow improvements in the survey parameters to be better reflected in the SFR constraints, and can be effectively utilized by future LIM experiments.