PDE Evans, 2nd edition, page 136
Assume $v : \mathbb{R} \times [0,\infty) \rightarrow \mathbb{R}$ is smooth, with compact support. We call $v$ a test function. Now multiply the PDE $u_t + F(u)_x=0$ by $v$ and integrate by parts:
\begin{align} 0&= \int_0^\infty \int_{-\infty}^{\infty} (u_t + F(u)_x)v \, dx dt \\ &= - \int_0^\infty \int_{-\infty}^{\infty} uv_t \, dx dt - \int_{-\infty}^{\infty} uv \, dx|_{t=0} - \int_0^\infty \int_{-\infty}^{\infty} F(u) v_x \, dx dt \end{align}
I am not understanding completely the "integrate by parts" process that was derived to achieve the expression in the last line. How can this expression be obtained? I tried applying the integration by parts formula (from the textbook's appendix):
Let $u,v \in C^1(\bar{U})$. Then for $i=1,\ldots,n$, $$\int_U u_{x_i} v \, dx = -\int_U uv_{x_i} \, dx + \int_{\partial U} uv \nu^i \, dS$$
I tried replacing $u_{x_i} \rightarrow u_t$ and $v_{x_i}=v_t$, and other similar substitutions.