Integration by parts in Sobolev space

Question

I'm looking for a reference of the following fact (if it is true...): if $u\in W^{1,1}(\Omega)$ and $v \in W^{1,\infty}(\Omega)$ ($\Omega$ a open subset of $\mathbb{R}^n$ ($n \ge 1$) with a regular boundary, one may suppose first that $\Omega$ is bounded) then we have $$ \int_\Omega \partial_{x_i} u v + \int_\Omega u \partial_{x_i} v = \int_{\partial \Omega} u v n_i $$ where $n_i$ is the $i$-th component of the exterior normal of $\Omega$. The same result is true when $u\in W^{1,p}(\Omega)$ and $v \in W^{1,q}(\Omega)$ with $1< p < \infty$ and $q$ is such that $1/p+1/q=1$ but I can't find a proof in this case.