# Evaluating an intergal

• May 8th 2014, 12:07 AM
pablohuds
Evaluating an intergal
Hello everyone,

I was reading something and the following equation turns up:

$\displaystyle \int^T_0 e^{-rT}(T-t)dt$ =$\displaystyle \frac{rT+e^{-rT}-1}{r^2}$

Where r is a constant. It has been a while since I last did integrals. When I tried to work it out I ended up with (expand the bracket and then integrate each term)

$\displaystyle \frac{e^{-rT}T^2}{2}$

I am fully willing to admit that I have made a stupid mistake somewhere as it has been a long time from when I last tried to solve one.

Thank you for any help!
• May 8th 2014, 03:35 AM
Prove It
Re: Evaluating an intergal
Is T a constant too?
• May 8th 2014, 04:13 AM
HallsofIvy
Re: Evaluating an intergal
Whether "T" is a constant or not since it is an upper limit on the integral it has to be treated as one.
So the integral is $\displaystyle Te^{-rT}\int_0^T dt- e^{-rt}\int_0^T tdt$. I get the same thing you do. Perhaps the equation is not saying that the integral on the left gives the result on the right, but that the integral on the left is, for some other reason, equal to the right side? That is, that $\displaystyle \frac{T^2e^{-rT}}{2}= \frac{rT- e^{-rT}- 1}{r^2}$? That doesn't seem quite right but it is the only way I can make sense of it. Could you give more information about what you were reading?