This paper proposes a Wirtinger-based integral inequality to improve the stability analysis of time-delay and sampled-data systems. The Jensen inequality, commonly used in such analyses, introduces conservatism. The proposed inequality, derived from Fourier theory, provides a more accurate lower bound for integral terms, reducing this conservatism. The inequality is shown to encompass the Jensen inequality and leads to tractable Linear Matrix Inequality (LMI) conditions. The paper applies this inequality to two cases: time-delay systems and sampled-data systems. For time-delay systems, the inequality is used to derive new stability criteria, which are expressed in terms of LMIs. For sampled-data systems, the inequality is applied to a functional that incorporates the state and its integral over the sampling interval, leading to improved stability conditions. Theoretical results are supported by numerical examples, demonstrating that the proposed inequality provides better bounds than the Jensen inequality. The results show that the new inequality can be used to derive more accurate stability conditions for both time-delay and sampled-data systems, with significant improvements over existing methods. The paper concludes that the Wirtinger-based integral inequality is a valuable tool for improving the analysis of time-delay and sampled-data systems.This paper proposes a Wirtinger-based integral inequality to improve the stability analysis of time-delay and sampled-data systems. The Jensen inequality, commonly used in such analyses, introduces conservatism. The proposed inequality, derived from Fourier theory, provides a more accurate lower bound for integral terms, reducing this conservatism. The inequality is shown to encompass the Jensen inequality and leads to tractable Linear Matrix Inequality (LMI) conditions. The paper applies this inequality to two cases: time-delay systems and sampled-data systems. For time-delay systems, the inequality is used to derive new stability criteria, which are expressed in terms of LMIs. For sampled-data systems, the inequality is applied to a functional that incorporates the state and its integral over the sampling interval, leading to improved stability conditions. Theoretical results are supported by numerical examples, demonstrating that the proposed inequality provides better bounds than the Jensen inequality. The results show that the new inequality can be used to derive more accurate stability conditions for both time-delay and sampled-data systems, with significant improvements over existing methods. The paper concludes that the Wirtinger-based integral inequality is a valuable tool for improving the analysis of time-delay and sampled-data systems.