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We prove necessary and sucient conditions for an arbitrary in- variant of braids with m double points to be the \mth derivative" of a braid invariant. We show that the \primary obstruction to integration" is the only obstruction. This gives a slight generalization of the existence theorem for Vassiliev invariants of braids. We give a direct proof by induction on m which works for invariants with values in any abelian group. We nd that to prove our theorem, we must show that every relation among four-term relations satises a certain geometric condition. To nd the relations among relations we show that H1 of a variant of Kontsevich's graph complex vanishes. We discuss related open questions for invariants of links and other things.

Integration of singular braid invariants and graph cohomology