  Null Lagrangians and their gauge functions are derived for given standard and
non-standard Lagrangians. The obtained standard null Lagrangians generalize
those previously found but the non-standard null Lagrangians are new. The gauge
functions are used to make the action invariant and introduce the exact null
Lagrangians, which form a new family of null Lagrangians. The conditions
required for the action to be invariant are derived for all null Lagrangians
presented in this paper, thus, making them exact. As a specific application,
the exact null Lagrangians are derived for a second-order ordinary differential
equation. The application shows that the exact null Lagrangians can be obtained
for any ordinary differential equation with known Lagrangian.
