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The Total Monte Carlo (TMC) technique has proven to be a powerful tool to propagate uncertainties in nuclear data to the uncertainty in macroscopic quantities, such as neutron fluxes at detector positions and the criticality of reactor cores. Nuclear data uncertainties can be used to create self-consistent sets of cross-sections. Each set contains files generated by variations of nuclear model parameters to properly fit the model to the nuclear data, accounting for their uncertainty. These files are called random files. The random files reflect the covariances of the nuclear data due to the uncertainties of the nuclear physics model parameters. TMC uses particle transport codes, such as MCNP, to transport particles through arbitrarily complex geometries. Each set of random files is used in a separate transport code run. This allows for the propagation of uncertainties in nuclear data, which otherwise could be hard to account for in the transport codes. However, particle transport techniques are well-known to be computationally expensive. The Half Monte Carlo (HMC) technique uses the random files of the TMC technique but does not rely on transport codes to propagate the uncertainties of nuclear data to the uncertainty of the sought macroscopic quantity. Instead, it uses pre-calculated sensitivity matrices to calculate the difference in a macroscopic quantity, given the difference of the random files relative to the best estimate of the nuclear data evaluation. In this work, we demonstrate how to use the HMC technique to calculate the uncertainty of macroscopic quantities in integral experiments for a set of random files relative to the best nuclear data evaluation. In this paper, we demonstrate how HMC can be used to incorporate integral experiments into an automated nuclear data evaluation. After applying the Bayesian Monte Carlo method in conjunction with the HMC technique and random files of uranium-235 from the TENDL library on the Godiva experiment, we conclude that the HMC technique gives similar results to that of the TMC technique: the mean value and the standard deviation of ∆keff is -6.30 pcm and 1220 pcm, respectively.