How to show $\int_\Omega u \partial_i \zeta = - \int_\Omega \partial_i u\zeta$ for compactly supported $\zeta$?

Question

Let $\Omega \subset \mathbb{R}^n$ be an open subset, $u \in \mathscr{C}^1(\Omega)$ and $\zeta \in \mathscr{C}_c^\infty(\Omega)$ a compactly supported smooth function. I want so show that for $i = 1, \dots, n$, $$\int_\Omega u \partial_i \zeta = - \int_\Omega \partial_i u \zeta.$$ If $n = 1$ and $\Omega$ is an interval, this follows from integration by parts, but I don´t know how to show the claim in higher dimensions.