Straightforward expansion of non-autonomous integrals for quasi-conservative systems with one degree of freedom

Quasi-conservative stationary systems with one degree of freedom are considered. Straightforward expansion of non-autonomous integrals for quasi-conservative systems is studied and analyticity of such integrals by small parameter is discussed. Method for constructing a set of non-autonomous integrals for quasi-conservative systems in action-angle variables is proposed. Criterion of closed orbits’ existence is obtained in terms of non-autonomous integrals. This criterion is used to estimate the number of limit cycles for one class of Lienard's equation.