  We develop the theory of Free Integro-Differential Algebras (FIDA) extending
the powerful technique of Free Differential Algebras constructed by D.
Sullivan. We extend the analysis beyond the superforms to integral- and
pseudo-forms used in supergeometry. It is shown that there are novel structures
that might open the road to a deeper understanding of the geometry of
supergravity. We apply the technique to some models as an illustration and we
provide a complete analysis for D=11 supergravity. There, it is shown how the
Hodge star operator for supermanifolds can be used to analyze the set of
cocycles and to build the corresponding FIDA. A new integral form emerges which
plays the role of the truly dual to 4-form $F^{(4)}$ and we propose a new
variational principle on supermanifolds.
