Integral forms are characteristic supergeometric objects that allow us to define a meaningful notion of integration on supermanifolds. In this talk, I will introduce a double complex of non-commutative sheaves that relates integral forms to the more customary notion of differential forms. I will then discuss how this framework specializes to so-called cotangent bundle supermanifolds, which are relevant to odd symplectic geometry and BV theory. If time permits, I will explain how the geometry of forms is related to the problem of splitting a complex supermanifold in this particular setting.