This article develops a new time integration family for second-order dynamic equations. A combination of the trapezoidal rule and higher-order Newton backward extrapolation functions are utilized in the formulation. Five members of the suggested family are extensively studied in this article. Most members of the presented time integration family are new. The stability and accuracy of the proposed time integration schemes are investigated by solving some benchmark problems. Numerical results are checked and compared with well-known strategies. The findings of the article show the efficiency, accuracy and robustness of the suggested techniques.