  In this paper we have chosen to work with two different approaches to solving
the inverse problem of the calculus of variation. The first approach is based
on an integral representation of the Lagrangian function that uses the first
integral of the equation of motion while the second one relies on a
generalization of the well known Noether's theorem and constructs the
Lagrangian directly from the equation of motion. As an application of the
integral representation of the Lagrangian function we first provide some useful
remarks for the Lagrangian of the modified Emden-type equation and then obtain
results for Lagrangian functions of (i) cubic-quintic Duffing oscillator, (ii)
Li\'{e}nard-type oscillator and (iii) Mathews-Lakshmanan oscillator. As with
the modified Emden-type equation these oscillators were found to be
characterized by nonstandard Lagrangians except that one could also assign a
standard Lagrangian to the Duffing oscillator. We used the second approach to
find indirect analytic (Lagrangian) representation for three velocity-dependent
equations for (iv) Abraham-Lorentz oscillator, (v) Lorentz oscillator and (vi)
Van der Pol oscillator. For each of the dynamical systems from (i)-(vi) we
calculated the result for Jacobi integral and thereby provided a method to
obtain the Hamiltonian function without taking recourse to the use of the
so-called Legendre transformation.
