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What is this integral

  1. Jul 2, 2014 #1
    What is this integral
    [itex]\int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{b}dx[/itex]
    where a and b are constants.
     
  2. jcsd
  3. Jul 2, 2014 #2

    pasmith

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    Homework Helper

    The substitution [itex]ax = \sinh t[/itex] yields [tex]
    \int \left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^b\,dx = \int \left(\frac{t}{\sinh t}\right)^b \frac{\cosh t}{a}\,dt \\
    = \left[ \frac{1}{a(1-b)}\frac{t^b}{(\sinh t)^{b-1}}\right]
    + \frac{b}{a(b - 1)} \int \left(\frac{t}{\sinh t}\right)^{b-1}\,dt \\
    [/tex] on integration by parts. Unfortunately that seems to be as far as one can get.
     
  4. Jul 9, 2014 #3

    Philip Wood

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    Gold Member

    The wonderful Wolfram online integrator can't do it, so there's not much hope...
     
  5. Jul 9, 2014 #4
    I confirm, Mathematica replies: "no result found in terms of standard mathematical functions" which is true in most cases.
     
  6. Jul 31, 2014 #5

    TheDemx27

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    Gold Member

    Just starting with Mathematica, I type in:
    Code (Text):
    Integrate[((ArcSinh[a * x])/ a * x)^b, x]
    and I get out:
    Code (Text):
    \[Integral]((x ArcSinh[a x])/a)^b \[DifferentialD]x
    Is there some reason I am getting a different output?
     
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