= Solution
If a <test function> $\varphi$ has integral zero, then
$$
\psi(x)=\int_{-\infty}^x\varphi(t)\,dt
$$
is another compactly supported smooth test function with $\psi'=\varphi$. Hence $v'=0$ implies $\langle v,\varphi\rangle=-\langle v',\psi\rangle=0$ for every such $\varphi$.
Choose a test function $\eta$ with integral one. For arbitrary $\varphi$, the function $\varphi-\eta\int\varphi$ has zero integral, so
$$
\boxed{\langle v,\varphi\rangle=\langle v,\eta\rangle\int_{\mathbb R}\varphi(x)\,dx.}
$$
Thus $v$ is the constant distribution $\langle v,\eta\rangle$. This proves <a distribution with zero derivative is constant>.
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