In this paper, we present the convergence behavior of a modified Newton's method based on Simpson integral rule. The convergence properties of this method for solving equations which have simple roots have been discussed and it has been shown that it converges cubically to simple roots. The values of the corresponding asymptotic error constants of convergence are determined. Theoretical results have been verified on the relevant numerical problems. In addition, we also point out that if combined with other numerical methods such as improved Ostrowski's method, the SN method can be more efficient.