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An integral by parts problem - please advise

  1. Mar 25, 2012 #1
    1. The problem statement, all variables and given/known data

    The function is increasing and has a inverse f^-1
    Also assume f′is continuous and f'(x) > 0 over the state interval of integration [a,b]

    PLEASE NOTE! a is lower limit, b is upper limit (same for alpha and beta symbol later on)

    Used integration by parts to show that:
    [itex]\int f(x)dx=bf(b)−af(a)−∫ba xf′(x)dx[/itex]

    Then if y = f(x), then the following is true (used the substitution rule)

    [itex] \int _a ^b f(x) \: dx = bf(b) - af(a) - \int _{f(a)} ^{f(b)} f ^{-1} (y) \: dy [/itex]
    2. Relevant equations

    The question: Now, I need to show that if α = f(a), and β = f(b), then

    [itex]∫βαf^-1(x)dx=β f^-1(β)−α f^-1(α)−∫f^-1(β)f^-1(α)f(x)dx[/itex]


    3. The attempt at a solution

    trying to fix the problem's display. B should be the upper limit, a should be lower limit when next to an integral sign

    Im lost. I'm sure its something relatively easy to do, maybe another substitution or relevant manipulation?

    Help appreciated!
     
    Last edited: Mar 25, 2012
  2. jcsd
  3. Mar 25, 2012 #2

    LCKurtz

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    It isn't true. Try f(x) = x, a = 1, b = 2.
     
  4. Mar 25, 2012 #3

    SammyS

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    You have a few (apparent) typos, which I attempted to fix.

    You have worked out most of the result.

    (To view the LaTeX code, right click the desired expression and choose "Show Math As: TEX Commands".)
     
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