In this work we establish a Stokes-type integral equality for scalarly essentially integrable forms on an orientable smooth manifold with values in the locally convex linear space $\langle B(G),\sigma(B(G),\mathcal{N})\rangle$, where $G$ is a complex Banach space and $\mathcal{N}$ is a suitable linear subspace of the norm dual of $B(G)$. This result widely extends the Newton-Leibnitz-type equality stated in one of our previous articles. To obtain our equality we generalize the main result of those articles, and employ the Stokes theorem for smooth locally convex vector-valued forms established there. Two facts are remarkable. First, the forms integrated involved in the equality are functions of a possibly unbounded scalar-type spectral operator in $G$. Secondly, these forms need not be smooth nor even continuously differentiable.