Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering the general case of a linearization problem via certain nonlocal transformations. In addition, we show that each equation from the linearizable family admits a transcendental first integral and study particular cases when this first integral is autonomous or rational. Thus, as a byproduct of solving this linearization problem we obtain a classification of second-order differential equations admitting a certain transcendental first integral. To demonstrate effectiveness of our approach, we consider several examples of autonomous and non-autonomous second order differential equations, including generalizations of the Duffing and Van der Pol oscillators, and construct their first integrals and general solutions. We also show that the corresponding first integrals can be used for finding periodic solutions, including limit cycles, of the considered equations.

非线性二阶常微分方程在科学的各个领域都很常见，例如物理学，力学和生物学。在这里，通过考虑通过某些非局部变换的线性化问题的一般情况，我们提供了一个新的可积分二阶常微分方程族。此外，我们证明了线性化族中的每个方程都承认先验的第一积分，并研究了该第一积分是自治的还是有理的特殊情况。因此，作为解决此线性化问题的副产品，我们获得了包含某个先验第一积分的二阶微分方程的分类。为了证明我们方法的有效性，我们考虑自治和非自治二阶微分方程的几个例子，包括Duffing和Van der Pol振荡器的推广，并构造它们的第一个积分和一般解。我们还表明，相应的第一积分可用于查找所考虑方程的周期解，包括极限环。