  A log symplectic manifold is a Poisson manifold which is generically
nondegenerate. We develop two methods for constructing the symplectic groupoids
of log symplectic manifolds. The first is a blow-up construction, corresponding
to the notion of an elementary modification of a Lie algebroid along a
subalgebroid. The second is a gluing construction, whereby groupoids defined on
the open sets of an appropriate cover may be combined to obtain global
integrations.
  This allows us to classify all Hausdorff symplectic groupoids of log
symplectic manifolds in a combinatorial fashion, in terms of a certain graph of
fundamental groups associated to the manifold. Using the same ideas, and as a
first step, we also construct and classify the groupoids integrating the Lie
algebroid of vector fields tangent to a smooth hypersurface.
