Answer
$$PV=\frac{R}{r}\left(1-e^{-r T}\right)$$
Work Step by Step
Since $$ R(t) =R $$
Then
\begin{align*}
P V&=\int_{0}^{T} R(t) e^{-r t} d t\\
&=\int_{0}^{T} R e^{-r t} d t\\
&=\left.\frac{R}{-r} e^{-r t}\right|_{0} ^{T}\\
&=\frac{R}{-r}\left(e^{-r T}-e^{0}\right)\\
&=\frac{R}{r}\left(1-e^{-r T}\right)
\end{align*}
