We here extend to integral constitutive equations of the Doi-Edwards type the analysis of shear banding instabilities during shear startup recently performed by Moorcroft and Fielding [J. Rheol. 58, 103-147 (2014)]. Shear banding is the instability encountered in simple shear flows, whereby different shear rates coexist along the sample thickness, so that the velocity profile departs from the usually assumed linear profile. Results of both linear and nonlinear stability analysis mostly confirm those obtained by Moorcroft and Fielding [J. Rheol. 58, 103-147 (2014)] with the Rolie-Poly equation [Likhtman, A. E., and R. S. Graham, J. Non-Newtonian Fluid Mechan. 114, 1-12 (2003)] (which is an approximate differential counterpart of the Doi-Edwards integral one "[The Theory of Polymer Dynamics (Clarendon, Oxford, 1986)]"), but some significant differences are also found. Features that coincide with those predicted by the Rolie-Poly equation include: (i) For the nonmonotonic flow curve of the original Doi-Edwards equation, the viscous instability classically predicted in the decreasing part of the flow curve is preceded during startup by an elastic instability in the region of the stress overshoot; (ii) to the right of the flow-curve minimum, a region of metastability is found, whereas, to the left of the maximum, attempts at finding metastability have failed; and (iii) when convective constraint release (CCR) is accounted for, and a monotonic flow curve is obtained, transient banding instabilities of elastic origin in the shear-rate range of the quasiplateau region are again found close to the overshoot. On the other hand, the integral Doi-Edwards equation appears not to predict transient elastic instabilities to the left of the maximum of the flow curve, differently from what found by Moorcroft and Fielding [J. Rheol. 58, 103-147 (2014)] with the Rolie-Poly equation. Also, for the case where CCR generates a monotonic flow curve, the shear-rate range in the plateaulike region where elastic instabilities are found comes out much narrower than for the Rolie-Poly. All considered, the integral Doi-Edwards equation appears to predict fewer instabilities than its differential counterpart, which is perhaps more in line with the contradictory experimental results on banding of entangled polymers reported by different authors. (C) 2016 The Society of Rheology.