When can we affirm this equality about integral? Suppose we have $f(x) = \int G(x_{0},x_{1},...)\,dx_{0}\,dx_{1}\dotsb dx_{n}$
When can we affirm that $$df = G(x_{0},x_{1},...)? \tag1$$
Basically, I am having trouble to understand how to deal with differentials of functions, intuitively I thought that we can do that:
$$\delta f(x) = \int \sum\left(\frac{\partial G}{\partial dx_{i}}\right)\,dx_{0}\,dx_{1}
\dotsb dx_{n} \tag2$$
But I am not sure how $(2)$ reduces to $(1)$.

In some notations this is true; for example, the volume element can be written $dV=A \ du \ dv$ for some $A$. In the case of your integral, the proper way of dealing with it is probably $df=Gdx_0dx_1\cdots dx_{n-1}dx_n$, for which its usefulness depends on the context.
