In this work the spline-collocations and spline-quadratures methods for solving integral equations (IE) are developed and theoretical substantiated. Also, the Intelligent Support System (ISS_IE) for approximate solving of regular and weak-singular integral Fredholm and Volterra equations of second kind is developed and implement based on the following received results: – development of spline-collocations and spline-quadratures computational algorithms for approximate solving of regular and weak-singular integral Fredholm and Volterra equations of second kind; – some modification of traditional quadrature-interpolations computational algorithms in solving IE. These new computational algorithms are effective and better adapted to their way of theoretical substantiation. As an example, the computational algorithms of spline-quadratures method in solving IE Fredholm and Volterra of second kind; – theoretical substantiation in H¨older spaces and continuous functions space of developed computational algorithms; – implementation of degenerated kernel method for exact solving the Fredholm and Volterra IE of second kind; – establishment of principles and architecture of development ISS_IE for regular and weak-singular integral Fredholm and Volterra equations of second kind.Based on the provided cores for regular and weak-singular integral equations (more than 2000 types) two knowledge bases were created: * The base of prototypes of integral equation kernels (BPK_IE_COMP) for checking the sufficient compatibility conditions of integral equations; * The base of prototypes of kernels of regular and weak-singular integral equations (BPK_IE_COL) for solving by the spline-colocations method. Integral Equations Solver (IES) for regular and weak-singular integral equations has been developed. The obtained algorithms were tested on several examples, and was found to be quite effective.