  Noncommutative functions are graded functions between sets of square matrices
of all sizes over two vector spaces that respect direct sums and similarities.
They possess very strong regularity properties (reminiscent of the regularity
properties of usual analytic functions) and admit a good
difference-differential calculus. Noncommutative functions appear naturally in
a large variety of settings: noncommutative algebra, systems and control,
spectral theory, and free probability. Starting with pioneering work of J.L.
Taylor, the theory was further developed by D.-V. Voiculescu, and established
itself in recent years as a new and extremely active research area. The goal of
the present paper is to establish a noncommutative analog of the Frobenius
integrability theorem: we give necessary and sufficient conditions for higher
order free noncommutative functions to have an antiderivative.
