The work includes algorithms and substantiation theory in some Banach space of spline-collocation and spline-quadratures methods in solving the following integral equations of second kind: 1) Fredholm linear integral equations of second kind; 2) Volterra linear integral equations of second kind; 3) Fredholm linear weak-singular integral equations of second kind; 4) Volterra linear weak-singular integral equations of second kind. For proposed integral equations, new spline-collocation and splinequadratures algorithms are developed, such as (see Abstracts of CAIM2017, CAIM-2018 and CAIM-2019):1) spline-collocations algorithms for solving Fredholm and Volterra linear integral equations of second kind, which use as basic functions some convex and concave splines; 2) spline-quadratures algorithms for solving Fredholm and Volterra linear integral equations of second kind, which use as basic functions the same convex and concave splines; 3) spline-collocations algorithms for solving Fredholm and Volterra linear weak-singular integral equations of second kind, which use as basic functions the linear and some convex and concave splines; 4) spline-quadratures algorithms for solving Fredholm and Volterra linear weak-singular integral equations of second kind, which use as basic functions the linear and some nonlinear splines. Following, we have established sufficient conditions on compatibility and convergence of the developed computing algorithms in spaces of continuous functions and Holder spaces.