Present value of a continuous perpetuity

Statement

An investment generates a constant continuous cash flow of $8000\ \text{€}$ per year forever. With a continuous discount rate of $5\%$, the present value is $\displaystyle\int_{0}^{\infty} 8000\,e^{-0{.}05t}\,dt$. Compute it.

Solution

$\int_{0}^{\infty}8000\,e^{-0{.}05t}\,dt=8000\left[-\frac{1}{0{.}05}e^{-0{.}05t}\right]_{0}^{\infty}=8000\cdot\frac{1}{0{.}05}\,(1-0).$ $VA=8000\cdot 20=\boxed{160\,000\ \text{€}}.$