Quadratic optimal control problems have applications in various fields of science and engineering at the same time, they are relatively easy to solve, In addition, these problems are often taken as test examples that demonstrate the effectiveness of basic methods of optimal control theory. Many applications contain integrals, such as electrical circuits that contain capacitors, and it is not feasible in applications to convert integrals into differentials, especially if there are many integrals of many variables in the problem. In this paper, we first consider the general finite time horizon control problem for nonlinear linear integral systems, and then proceed to discuss the case for linear quadratic problem as well as we will construct an algorithm to find the exact solution of the quadratic optimal control problem for a system described by linear integral equations. As applications, this algorithm will be used to compute the optimal solution for single and coupled RC electrical circuits.