With the increasing penetration of distributed generations (DGs), the equations governing active distribution networks (ADNs) exhibit stronger nonlinearity and greater stiffness. Additionally, the uncertainties associated with DGs mean that ADNs face more frequent and diversified disturbances. The novel properties of ADNs exacerbate the instabilities and computational burdens on iterations of time-domain simulation when using traditional explicit and implicit integration algorithms. This article proposes a novel semi-implicit integration method incorporating an adaptive Jacobian matrix to solve the differential equations (DEs) governing ADNs, resulting in a non-iterative technique with good numerical stability. The proposed approach simultaneously combines the advantages of both explicit and implicit methods. Moreover, a parameter optimization strategy that comprehensively considers stability, efficiency, and accuracy conditions and an adaptive Jacobian matrix update strategy are developed to further improve the numerical performance of the proposed method. Finally, the proposed method is validated using a modified 33-node system and a practical 436-node distribution system. The simulation results demonstrate the prominent advantages of the proposed method in terms of stability and efficiency compared with the modified Euler and trapezoidal methods.