Thanks to the good tracking and calorimetric capabilities of a Liquid Argon Time Projection Chamber (LArTPC), the Micro Booster Neutrino Experiment (MicroBooNE) provides precise measurements of Î½-argon interactions at intermediate energies between O(100MeV) and O(2 GeV). These are needed to improve our understanding of neutrino-nucleus interactions and to mitigate the uncertainties in the Monte Carlo simulation which needs to predict the signals of future experiments like Deep Underground Neutrino Experiment (DUNE) with great precision. A drawback of the LArTPC technology is the rather long charge readout in the of O(1 ms). This means that a large amount of cosmic-induced particles will overlap the readout window for detectors placed on the earthâ€™s surface. To mitigate this problem in MicroBooNE, a Cosmic Ray Tagger (CRT) is built around the LArTPC. This work discusses the implementation of the CRT and shows its implication for a Î½Î¼ Charged Current (CC) inclusive analysis. With a timing capability in the O(2 ns) and a position resolution better than 2 cm, the CRT can reject cosmic background interactions without confinging the phase space of the selected neutrino interactions. On a newly developed Î½Î¼ CC inclusive analysis using MicroBooNE data taken between December 2017 and Summer 2018, the cosmic contribution could be halved while maintaining the efficiency of 55.6% at the same level with respect to previous analysis. This leads to a higher signal purity of 72.0% and generally reduced uncertainties. The total flux-integrated inclusive cross section evaluated in this analysis is Ïƒ = 0.770 Â± 0.005 (stat) Â± 0.113 (sys) Â· 10âˆ’38 cm2. A double differential result in bins of muon momentum and the polar angle with respect to the beam direction is presented as a forward-folded event rate to avoid model dependence to a large extend. A comparison of the result to different neutrino generators commonly used results in Ï‡2-values larger than 10 per degrees of freedom for all models used here indicating a general discrepancy between models and data for neutrino-argon interactions. This work shows an effective way to further remove cosmic background interactions for a LArTPC on the earth surface using the CRT and presents a double differential cross-section analysis for further improvement of our understanding of Î½-argon interactions.