This technical note is concerned with the problem of stability analysis for time-delay systems. A new series of integral inequalities to bound a single integral term is presented by introducing some free matrices, which produces tighter bounds than some existing ones. Similarly, based on orthogonal polynomials defined in integral inner spaces, new series of multiple integral inequalities are presented as well, which include the existing double ones. To show the effectiveness of the proposed inequalities, their applications to stability analysis of systems with discrete and distributed delays are provided with numerical examples.