This paper is concerned with the delay-dependent stability for the linear systems with a time-varying delay. To get a result in the form of LMI from a Lyapunov-Krasovskii functional, an integral inequality is necessary and Jensen inequality has been a most powerful inequality in the last few years. Recently, based on Wirtinger inequality, an improved integral inequality, encompassing Jensen inequality, was proposed and its application to the stability showed a quite improvement. In this paper, without using Wirtinger inequality, a further improved integral inequality in the form of infinite series is derived, and, based on this, a delay-dependent stability condition in the form of LMI is derived. Finally, its contribution on the stability criterion is shown by well-known two examples. (C) 2015 Elsevier Ltd. All rights reserved.