Studying calculus of variations from the book of Arfken's Mathematical Methods for physicists, reaching the part of the Euler Lagrange equations for functionals with several independent variables(section 22.2, page 1100 on the 7th edition), it is stated that an integration by parts over a multiple integral (triple, in this case) in order to "eliminate" the derivatives of the variation function $\eta$ is made, i.e.:
$$\frac{\partial J}{\partial \alpha}=\int\int\cdots\int\left(\frac{\partial f}{\partial y}\eta+\frac{\partial f}{\partial y_{x_1}}\eta_{x_1}+\cdots+\frac{\partial f}{\partial y_{x_i}}\eta_{x_i}+\cdots+\frac{\partial f}{\partial y_{x_n}}\eta_{x_n}\right)dx_1\cdots dx_n =0$$
Where $y_{x_{i}}=\partial y/\partial x_{i}$, and $\eta_{x_{i}}=\partial\eta /\partial x_{i}$.
On the other cases, where there is only one independent variable, they make a substitution of the terms with the derivatives of the variation functions $\eta_{x_{1}}$ as it follows, taking the i-th therm:
$$\frac{\partial}{\partial x_{i}}\left(\frac{\partial f}{\partial y_{x_i}}\eta\right)=\frac{\partial}{\partial x_{i}}\left(\frac{\partial f}{\partial y_{x_i}}\right)\eta+\frac{\partial f}{\partial y_{x_i}}\eta_{x_{i}}$$
$$\therefore \frac{\partial f}{\partial y_{x_i}}\eta_{x_{i}} = \frac{\partial}{\partial x_{i}}\left(\frac{\partial f}{\partial y_{x_i}}\eta\right) - \frac{\partial}{\partial x_{i}}\left(\frac{\partial f}{\partial y_{x_i}}\right)\eta$$
Where, if this was the case of just a single independent variable, the first term on the RHS would later be eliminated after it being integrated over the condition of the variation function $\eta$ being zero on the extreme points.
Considering only the i-th terms, replacing the latter expression into the functional:
$$\int\int\cdots\int\left(\frac{\partial f}{\partial y}\eta+\cdots+ \frac{\partial}{\partial x_{i}}\left(\frac{\partial f}{\partial y_{x_i}}\eta\right) - \frac{\partial}{\partial x_{i}}\left(\frac{\partial f}{\partial y_{x_i}}\right)\eta+\cdots\right)dx_1\cdots dx_n=0$$
Nonetheless, with several integrals on the functional, one can't distribute these over the sum of the terms(or can I), nor separate the integral using Fubini's theorem. ¿How should I proceed?
As a clarifying annotation, I take the partial derivative of the i-th term only with respect to the i-th independent variable $x_{i}$, because the resulting EL equation is:
$$\frac{\partial f}{\partial y}=\sum^n_{i=1}\frac{\partial}{\partial x_i}\left( \frac{\partial f}{\partial y_{x_i}} \right)$$