Integrating Using Partial Fractions

[Jay9313]

Homework Statement

This is an arc length problem in three dimensions. I was given the vector r(t)=<e^t, 1, t> from t=0 to t=1

Homework Equations

Arc Length= \int |\sqrt{r&#039;(t)}| dt from t₁ to t₂
where |\sqrt{r&#039;(t)}| is the magnitude of the derivative of the vector

The Attempt at a Solution

I took the derivative and got the magnitude and simplified it down to
∫ √(e^2t+1) dt
I then set u=e^2t+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.

 

[vela]

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

 

[Jay9313]

That's not really working at all

 

[vela]

What did you try?

 

[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.

 

[vela]

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:
$$\frac{v^2}{v^2-1} = \frac{(v^2-1)+1}{v^2-1} = 1 + \frac{1}{v^2-1}.$$