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Integrating Using Partial Fractions

  1. Mar 10, 2013 #1
    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= [itex]\int[/itex] |[itex]\sqrt{r'(t)}[/itex]| dt from t1 to t2
    where |[itex]\sqrt{r'(t)}[/itex]| 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:∫[itex]\frac{\sqrt{(u)}}{u-1}[/itex] du

    My professor said to use partial fractions from here, but I'm not sure how to do that.
     
  2. jcsd
  3. Mar 10, 2013 #2

    vela

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    Try one more substitution to get rid of the square root on top first. Then do partial fractions.
     
  4. Mar 10, 2013 #3
    That's not really working at all
     
  5. Mar 10, 2013 #4

    vela

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    What did you try?
     
  6. Mar 10, 2013 #5
    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.
     
  7. Mar 10, 2013 #6

    vela

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