In this paper, the weakly singular linear and nonlinear integro-differential equations are solved by using expansion method of Chebyshev polynomials of the first kind .The approximation solution of this equation is calculated in the form of a series which its components are computed easily .The existence and uniqueness of the solution and the convergence of the proposed method are proved. Numerical examples are studied to demonstrate the accuracy of the presented method. Keywords: China insurance industry, Volterra integral equations, Fredholm integral equations, Integro-differential equations, Singular integral equations, Chebyshev polynomials method
