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The current work investigates basic theory of finite integration method for first-order system of ordinary differential equations. We firstly investigate the error analysis of generalized finite integration methods which are constructed by ordinary linear approach. Then, the basic concepts of the error bounded theorems are discussed for a spectral meshless method derived from quadrature rule producing by radial point collocation method. By applying the finite integration methods for the first-order system of ordinary differential equation, we compare the accuracy residual error of the presented methods. The convergence of finite integration methods is proposed to confirm the standard criteria with theoretically accurate results. Finally, we extend the results for n-order system of ordinary differential equations.