Integration By Parts Problem

**Problem:**

During a surge in the demand for electricity, the rate, *r*, at which energy is used can be approximated by the following function where *t* is the time in hours and *b* is a positive constant.

$\displaystyle r(t) = te^{-bt}$

a.) (a) Find the total energy, *E*, used in the first *T* hours. Give your answer as a function of *b*.

**My Work:**

So I know that I want the integral of r(t) from 0 to T. I tried to find the integral and got:

$\displaystyle \frac{1}{b} e^{-bt}*(1-t)$

Is this right?

And also, I have NO IDEA what it means when the question says that it wants the answer **in terms of b. **"b" is a constant! How do you do that?! Is that like E(b), and if so, what would that look like?