We study the complexity of Banach space valued integration in the randomized setting. We are concerned with r-times continuously differentiable functions on the d-dimensional unit cube Q, with values in a Banach space X, and investigate the relation of the optimal convergence rate to the geometry of X. It turns out that the n-th minimal errors are bounded by cn −r/d−1+1/p if and only if X is of equal norm type p.