Improper Integral?
The present value of a perpetual stream of income that flows continually at the rate of P(t) dollars/year is given by the following equation where r is the interest rate compounded continuously.
PV = integral from infinity to 0 of P(t)e^(-rt)dt
Using this formula, find the present value of a perpetual net income stream that is generated at the rate P(t) = 13000 + 5500t. (Note: Your answer will contain the variable r.)
To be honest I don't even know where to start.
I wrote integral from infinity to 0 of 13000e^(-rt) + 5500te^(rt) dt.... but after that I have no clue. Any ideas?
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Answers & Comments
I am assuming that r is a positive constant.
Use integration by parts, with
u = 13000 + 5500t
du = 5500dt
dv = e^(-rt)
v = (-1/r)e^(-rt).
Recall that if u and v are functions of t,
integral from t=b (on top) to t=a (on bottom) of udv
= (uv at t=b) - (uv at t=a) - integral from t=b to t=a of vdu.
Therefore, keeping in mind that exponential decay outweighs polynomial growth, we have
integral from infinity to 0 of (13000 + 5500t)e^(-rt) dt
= [(13000 + 5500t)(-1/r)e^(-rt) at t=infinity] - [(13000 + 5500t)(-1/r)e^(-rt) at t=0]
- integral from infinity to 0 of (-1/r)e^(-rt)*5500dt
= 0 + 13000/r - {[5500(1/r^2)e^(-rt) at t=infinity] - [5500(1/r^2)e^(-rt) at t=0]}
= 0 + 13000/r - (0 - 5500/r^2)
= 13000/r + 5500/r^2.
Have a blessed, wonderful day!