  Until now, various methods have been demonstrated to solve the
Friedmann-Lama\'{\i}tre-Robertson-Walker (FLRW) equations in the spatially flat
$(k=0)$ cosmological model. In this study, in order to solve the field
equations of the spatially flat FLRW cosmological model in the presence of
$\Lambda$, a new method based on the invariants of the symmetry groups which we
called ISG-method, is presented. This method is based on the extended
Prelle-Singer (PS) method and it uses the Lie point symmetry,
$\lambda$-symmetry and Darboux polynomials (DPs). We employ this method to
extract systematically the two independent first integrals (or invariants) such
as $I_1 (a,\dot{a})=c_1$ and $I_2 (t,a,\dot{a})=c_2$ associated to the group of
the Lie point transformations keeping the Friedmann-Einstein dynamical equation
(DE), $\ddot{a}=\phi(t,a,\dot{a})$, invariant. The obtained solutions from
solving the DEs of the FLRW cosmological model by the ISG-method are explicitly
written, so that they are suitable for cosmological applications. Finally, as
an application of the solutions we look at the age of universe in the presence
of cosmological constant where the dominant matter of the universe is
considered to be fluid with the state parameter $w$. In this regard, we
calculate the age of universe when the dominant matter is dust.
