We consider a family of cubic Liénard oscillators with linear damping. Particular cases of this family of equations are abundant in various applications, including physics and biology. There are several approaches for studying integrability of the considered family of equations such as Lie point symmetries, algebraic integrability, linearizability conditions via various transformations and so on. Here we study integrability of these oscillators from two different points of view, namely, linearizability via nonlocal transformations and the Darboux theory of integrability. With the help of these approaches we find two completely integrable cases of the studied equation. Moreover, we demonstrate that the equations under consideration have a generalized Darboux first integral of a certain form if and only if they are linearizable.
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