Let $\Omega \subset \mathbb{R}^N$ is open , $u\in C^1(\Omega)$.
How can we prove that : $i\in \{1,...,N\}$
$$\int_{\Omega}\frac{\partial u}{\partial x_i}(x)v(x)dx=-\int_{\Omega}u(x)\frac{\partial v}{\partial x_i}(x)dx,\forall v \in C_c^1(\Omega).$$
Note : We already know that Green theorem (divergence theorem) can be use if $\Omega$ open , bound has piecewise smooth boundary and the $\Omega$ we have is no information about boundary.
Since $v$ has compact support, you can replace $\Omega$ by a ball of sufficiently large radius $R$, i.e.
$$\int_\Omega u_{x_i}v \, dx = \int_{B(0,R)} u_{x_i}v \, dx$$
and
$$\int_\Omega uv_{x_i} \, dx = \int_{B(0,R)} uv_{x_i} \, dx$$
provided the support of $v$ is contained in $B(0,R)$. Both $u$ and $v$ are defined to be zero in $B(0,R)\setminus \Omega$. The ball has a smooth boundary so you can use the form of Green's Theorem that you are familiar with.
Basically, if one of the functions is compactly supported, then you can ignore any boundary regularity issues.