Present value of a ten-year continuous flow

Statement

A project yields a continuous cash flow of $5000\ \text{€}$ per year for $10$ years. With continuous discounting at $4\%$, the present value is $\displaystyle\int_{0}^{10} 5000\,e^{-0{.}04t}\,dt$. Compute it.

Solution

$\int_{0}^{10}5000\,e^{-0{.}04t}\,dt=5000\left[-\frac{1}{0{.}04}e^{-0{.}04t}\right]_{0}^{10}=125000\,(1-e^{-0{.}4}).$ $VA=125000\,(1-0{.}670320)=125000\cdot 0{.}329680\approx\boxed{41\,210\ \text{€}}.$