A first integral is a conserved quantity associated with a coordinate or the independent variable.
A first integral associated with a coordinate x[t], y[t], ... is returned if f is independent of that coordinate, although f may contain derivatives of the coordinate. Such coordinates are typically called cyclic or ignorable coordinates.
A first integral associated with the independent variable t is returned if f is independent of t and does not contain any second or higher derivatives of the coordinates.
In mechanics, a first integral corresponding to a coordinate is typically associated with conservation of momentum, and a first integral corresponding to the independent variable is typically associated with conservation of energy.
FirstIntegrals returns a list of rules of the form FirstIntegral[u]->c, where u may be either the coordinates x, y, ... or the independent variable t, and c is the conserved quantity.