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Compute the present value of an investment that generates income at a rate of $5000 t e^{0.01 t}$ dollars per year forever, assuming an interest rate of $6 \%$.

The present value of that investment will be $\$ 2,000,000$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

TECHNIQUES OF INTEGRATION

Section 6

Improper Integrals

Integration

Integration Techniques

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and this problem going to take a look at, um, a rate of 5000 t e to the 0.1 T dollars per year that's invested forever at a rate of 6%. So are present value that we're trying to analyze. Here is the integral from zero to infinity of 5000 Tee hee hee to the 0.0 won t times e to the power of negative 0.6 t d t. And before we analyze this improper in a girl that simplifies we have 5000 t e to the negative 0.5 t d. T. The anti derivative of this one is gonna require integration by parts. So let's take a look at just this integral of t e to the negative 0.5 t. And, um, we'll deal with 5000 little bit later. So let's let you equal t, which gives us d'you equal d t and D be equal to e toothy negative through the power date of 0.5 t d t. And that gives us V equal to e to the negative 0.5 t divided by negative. 0.5 Okay. Our integration by parts formula then gives us the anti derivative um t multiplied by e to the negative 0.5 t divided by negative 0.5 minus. And then we're gonna multiply the times, do you? That ends up giving us a plus and then the inner girl of E to the negative 0.5 t divided by 0.5 times D t. And then our anti derivative of this. We'll get the tee times each and negatives your points, your five t and we divide that by negative 0.5 And then we do the anti derivative of this a second time, which gives us er minus e to the negative 0.5 t divided by 0.25 OK, so that's our anti derivative. And so let's use that now back into our improper interval. And this time we'll make sure we include 10,000 times the limit as t goes to infinity of this anti derivative we just found. And then it's gonna be evaluated from zero T. And so if you know if we let t go to infinity here, Uh, and we try to simplify this down. What we're gonna get is 5000 multiplied by one over 0.25 and that value in a dollar amount is $2 million.

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