So I need to show that, given $u,v \in W^{1,2}(\Bbb{R})$, then \begin{equation} \int_\Bbb{R}{uv'}dx=-\int_\Bbb{R}{u'v}dx \end{equation} My attempt has been this:
If $u,v \in W^{1,2}(\Bbb{R}) \Rightarrow uv\in W^{1,2}(\Bbb{R})$ with weak derivative $(uv)'=u'v+uv'$. Let $\varphi \in C^\infty_c(\Bbb{R})$ be a test function, then: $$\int_\Bbb{R}{(uv)\varphi'}dx=-\int_\Bbb{R}{(u'v+uv')\varphi}dx$$ $$\int_\Bbb{R}{(uv)\varphi'}dx=-\int_\Bbb{R}{u'v}\varphi dx-\int_\mathbb{R}{uv'\varphi}dx$$
As this equality holds for every test function $\varphi \in C^\infty_c(\Bbb{R})$, I think that I need to find an adequate test function that yields the thesis.
As $u,v\in L_ {loc}^1(\Bbb{R})$ and $u',v'\in L^2(\Bbb{R})$, I suppose that said test function has something to do with the support of $u,v,u',v'$.
Thanks in advance.