Background <p>The nonlinear vibrations with damping have been extensively studied with many methods for exact and approximate solutions, and the scope and applications of such nonlinear problems have been changed rapidly with late considerations of flexible structures and soft matters. Recently, it is also found that the nonlinear vibrations can be effectively analyzed by the Extended Galerkin Method (EGM) through the integration over time in one period of vibrations for both free and forced vibration problems;</p> Purpose <p>This paper aims to present a detailed analytical procedure that highlights the effectiveness of the EGM in the consideration of damping effect with integrations over infinity of time in connection with the exponentially decaying factor;</p> Methods&#xa0; <p>The EGM was extended with the integration interval expanded to infinite numbers of periods. An exponential decaying term was included in the trial function, leading to algebraic equations of unknown parameters from the weighted integrations over the infinite numbers of periods, or infinity in the evaluation;</p> Results <p>The approximate results obtained using the EGM are consistent with those from other established methods, thereby validating the applicability, accuracy, and reliability of the EGM for analyses of nonlinear vibrations;</p> Conclusions <p>The EGM proves effective for dealing with complex nonlinear vibration problems, offering advantages in simplifying the computational process and guaranteeing accurate results, confirming its value as a powerful analytical tool for nonlinear vibration analysis.</p>