Answer
See the detailed answer below.
Work Step by Step
First, we need to sketch this problem as shown below.
We know that
$$\Delta V=V_f-V_i=-\int_i^fE_rdr$$
where $v_f=V_r$ and $V_i=V_R=V_0$
$$ V_r-V_0=-\int_R^rE_rdr$$
where $E_r=\dfrac{\lambda}{2\pi \epsilon_0 r}$
$$ V_r-V_0=-\int_R^r \dfrac{\lambda}{2\pi \epsilon_0 r}dr$$
$$ V_r=V_0 -\dfrac{\lambda}{2\pi \epsilon_0}\int_R^r \dfrac{1}{ r}dr$$
$$ V_r=V_0 -\dfrac{\lambda}{2\pi \epsilon_0}\ln r\bigg|_R^r $$
$$\boxed{ V_r=V_0 -\dfrac{\lambda}{2\pi \epsilon_0}\ln\left[\dfrac{r}{R}\right] }$$
