The coordinated solution of AC optimal power flow (ACOPF) in the integrated transmission and distribution grid utilizes controllable energy resources in distribution grids for introducing additional operational economy and security benefits of power grids. However, it is rather impractical to compute the ACOPF centrally considering the information privacy in the operation of transmission and distribution grids. Due to the large capacity of distributed generators integrated to distribution grids, existing decentralized methods may encounter numerical problems and fail to converge when nonlinear ACOPF models are applied. In this paper, we propose a decentralized method to solve the ACOPF for integrated transmission and distribution grids. The proposed ACOPF model is characterized through polar-coordinate equations and branch flow equations for transmission grids and distribution grids respectively. A distribution-cost-correction framework is developed for a fast-convergent solution which is based on approximated distribution cost functions. A rigorous proof is provided for convergence and numerical simulations are analyzed for standard IEEE systems which demonstrate the advantages of the coordinated approach in reducing generation dispatch costs and mitigating overvoltage problems. The paper validates through simulations that the accuracy, computational efficiency, and scalability of the proposed approach are superior to those of traditional methods.