Noether's paper:
An integral is invariant under $$ y^i = x^i + \Delta x^i , \ \ v^i(y) = u^i + \Delta u^i, $$ whenever $$ \int f\left(y,v(y),\dfrac{\partial v}{\partial y}\right)dy =\int f\left(x,u(x),\dfrac{\partial u}{\partial x}\right)dx $$ holds. Fairly intuitive. Noether, later on, states that the integral over $dy$ can be written as
\begin{equation} \int f\left(y,v(y),\dfrac{\partial v}{\partial y}\right)dy = \int f\left(x,v(x),\dfrac{\partial v}{\partial x}\right) \ dx + \int \text{Div}(f\Delta x) \ dx. \end{equation}
The variation in $x$, if it is performed, it induces changes also in
$$
v(y)=v(x+\Delta x)=v(x)+\dfrac{\partial v}{\partial x^a}\Delta x^a + \mathcal{O}(\Delta x^2)
$$
and likewise the $\partial v/\partial y$ terms in the $f$, both of which are missing from the central equation above. Is she using the equations of motion?