An investigation has been conducted on the flaw of delay-dependent stability found in time-vary delay systems, followed by the selection of a suitable Lyapunov function, with a modification of some integral terms in Lyapunov function and the addition of an integral term. The improved results can be obtained by combining the relaxation conditions of partial matrices based on free matrix integral inequalities with the second-order Bessel-Legendre integral inequalities. The conservativeness and superiority of this proposed method can be verified by numerical examples.