This note proposes a systematic and more generic method to construct general bounded integral control. It is established by defining three new function sets and citing two function sets to construct three kinds of general bounded integral control actions and integrators, resorting to a universal strategy to transform ordinary control into general integral control and adopting Lyapunov method to analyze the stability of the closed-loop system. A universal theorem to ensure regionally as well as semi-globally asymptotic stability is provided in terms of some bounded information, and even does not need exact knowledge of Lyapunov function. Its one feature is that the indispensable element used to construct the general integrator can be taken as any integrable function, which satisfies Lipschitz condition and the self excited integral dynamic is asymptotically stable. Another feature is that the method to construct general bounded integral control action is extended to a wider function set. Based on this method, the control engineers not only can choose the most appropriate control law in hand but also have more freedom to construct the bounded integral control actions and integrators, and then a high performance integral controller is more easily found. As a result, the generalization of the bounded integral control is achieved.