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This paper deals with the problem of the asymptotical stability for a class of linear systems with time delay. First, a series of integral inequalities based on quadratic term are formulated by combining Leibniz-Newton formula. Next, using Lyapunov-Krasovskii functional method, the sufficient conditions of delay-dependent stability based on linear matrix inequality (LMI) are derived to ensure the linear systems with time delay are asymptotically stable. Last, some examples are given to illustrate that the new methods are more effective than the present methods and the delay bounds obtained in this paper are of less conservative.