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In recent years methods for the integration of Poisson manifolds and of Liealgebroids have been proposed, the latter being usually presented as ageneralization of the former. In this note it is shown that the latter methodis actually related to (and may be derived from) a particular case of theformer if one regards dual of Lie algebroids as special Poisson manifolds. Thecore of the proof is the fact, discussed in the second part of this note, thatcoisotropic submanifolds of a (twisted) Poisson manifold are in one-to-onecorrespondence with possibly singular Lagrangian subgroupoids ofsource-simply-connected (twisted) symplectic groupoids.Comment: 17 pages. Replacement: 18 pages, minor corrections, references added.  To appear in Lett. Math. Phy

On the Integration of Poisson Manifolds, Lie Algebroids, and Coisotropic Submanifolds