This paper compares different integrals of scalar functions \(f\) with respect to a strongly bounded, finitely additive, Banach space valued measure \(m\) on a \(\sigma\) algebra of sets. The version of the \(m\)-integral used here takes \(\int_{(\cdot)}f dm\) to be the setwise limit of \(\int_{(\cdot)}f_n dm\) whenever the limit exists for some sequence of simple functions \(\{f_n\}\) that converges in measure to \(f\) with respect to a control measure for \(m.\) The second integral considered is a gauge integral whose definition follows a generalization of the McShane integral due to \textit{D. H. Fremlin} [Ill. J. Math. 39, No. 1, 39-67 (1995; Zbl 0810.28006)]. The main result is that these two integrals are equivalent in the given setting. In the case that the Banach space is separable and \(m\) is the indefinite integral of a bounded integrable function with respect to a control measure, it is noted that the equivalence extends to the so-called monotone or Choquet integral which takes the form \(\int_0^{\infty} m([f>t]) dt\) whenever \(f\) is nonnegative.