# Integrating Using Partial Fractions

1. Mar 10, 2013

### Jay9313

1. The problem statement, all variables and given/known data
This is an arc length problem in three dimensions. I was given the vector r(t)=<et, 1, t> from t=0 to t=1

2. Relevant equations
Arc Length= $\int$ |$\sqrt{r'(t)}$| dt from t1 to t2
where |$\sqrt{r'(t)}$| is the magnitude of the derivative of the vector

3. The attempt at a solution

I took the derivative and got the magnitude and simplified it down to
∫ √(e2t+1) dt
I then set u=e2t+1
I then simplified and substituted until I got to:∫$\frac{\sqrt{(u)}}{u-1}$ du

My professor said to use partial fractions from here, but I'm not sure how to do that.

2. Mar 10, 2013

### vela

Staff Emeritus
Try one more substitution to get rid of the square root on top first. Then do partial fractions.

3. Mar 10, 2013

### Jay9313

That's not really working at all

4. Mar 10, 2013

### vela

Staff Emeritus
What did you try?

5. Mar 10, 2013

### Jay9313

I split it up, and it made it worse, and I had to do long division, but I got an answer. It's nasty, but I got an answer.

6. Mar 10, 2013

### vela

Staff Emeritus
Presumably, you ended up with something like $\frac{v^2}{v^2-1}$ after the substitution. A good technique to avoid doing long division is to add and subtract judiciously:
$$\frac{v^2}{v^2-1} = \frac{(v^2-1)+1}{v^2-1} = 1 + \frac{1}{v^2-1}.$$