  We consider the problem of integration of L_\infty-algebroids (differential
graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan
simplicial manifold (Fr\'echet or Banach) whose points are solutions of a
(generalized) Maurer-Cartan equation. The main analytic trick in our work is an
integral transformation sending the solutions of the Maurer-Cartan equation to
closed differential forms. Following ideas of Ezra Getzler we then impose a
gauge condition which cuts out a finite-dimensional simplicial submanifold.
This "smaller" simplicial manifold is (the nerve of) a local Lie k-groupoid.
The gauge condition can be imposed only locally in the base of the
L_\infty-algebroid; the resulting local k-groupoids glue up to a coherent
homotopy, i.e. we get a homotopy coherent diagram from the nerve of a good
cover of the base to the (simplicial) category of local Lie k-groupoids.
  Finally we show that a m-symplectic differential graded manifold integrates
to a local m-symplectic Lie k-groupoid; globally these assemble to form an
A_\infty-functor. As a particular case for m=2 we obtain integration of Courant
algebroids.
