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Integration problem?

  1. Nov 7, 2009 #1
    1. The problem statement, all variables and given/known data
    ∫ 1/r³ dr/dt


    2. Relevant equations

    ∫ xa dx = x(a+1)/(a+1)

    3. The attempt at a solution
    I have no clue how to solve it like this. I don't have the equation for r in terms of t, so I can't just substitute. How would I do it?
     
  2. jcsd
  3. Nov 7, 2009 #2

    lanedance

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    Homework Helper

    is that how the question is actually written & what are you integrating with respect to?

    it doesn't really make sense how you have written it, do you mean:
    [tex] \int dt (\frac{1}{r(t)^3}\frac{dr}{dt}) [/tex]

    if so have a think about how chain rule differentiation works & how you could try & reverse it using FTC...
     
  4. Nov 7, 2009 #3
    OK if it were like that, wouldn't the dt's just divide out allowing you to integrate with respect to r?
    ∫ dt(1/r³ dr/dt)
    ∫ 1/r³ dr
    -1/(2r²) + C
     
    Last edited: Nov 7, 2009
  5. Nov 7, 2009 #4

    lanedance

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    you can, but its a bit of an abuse of notation, athough it gives the same answer, a better way to think of it is to write:

    [tex] \frac{d}{dt} (-\frac{1}{2r(t)^2}) = \frac{1}{r(t)^3}\frac{dr}{dt}[/tex]

    then the intergal becomes
    [tex] \int dt (\frac{d}{dt} (-\frac{1}{2r(t)^2})) [/tex]

    so by FTC, the anti-derivative is
    [tex] = (-\frac{1}{2r(t)^2}) +C[/tex]

    though the question was to you - how is it written in the actual problem?
     
  6. Nov 7, 2009 #5
    It's not on a worksheet or anything; I was just messing with a particular physical situation, and I came to something in that form that I needed to integrate.
     
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