An integration theory for metric-space-valued functions is presented. Following a Bochner-type construction, the free integral of a strongly measurable function is defined as an element of the Lipschitz-free space F(M). The main properties of this integral are established, including duality with Lipschitz functions and the study of the resulting complete metric space of integrable functions. In the Banach-valued case, the free integral admits a natural decomposition that generalises the Bochner integral. On the geometric side, the free integral is shown to always produce convex integrals of molecules, which leads to results on the extremal structure of the unit ball of F(M). A final detailed example on a finite metric tree illustrates the theory.