What is this integral

  • Thread starter JulieK
  • Start date
  • #1
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What is this integral
[itex]\int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{b}dx[/itex]
where a and b are constants.
 

Answers and Replies

  • #2
pasmith
Homework Helper
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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.
 
  • #3
Philip Wood
Gold Member
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78
The wonderful Wolfram online integrator can't do it, so there's not much hope...
 
  • #4
77
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I confirm, Mathematica replies: "no result found in terms of standard mathematical functions" which is true in most cases.
 
  • #5
TheDemx27
Gold Member
170
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Just starting with Mathematica, I type in:
Code:
Integrate[((ArcSinh[a * x])/ a * x)^b, x]
and I get out:
Code:
\[Integral]((x ArcSinh[a x])/a)^b \[DifferentialD]x
Is there some reason I am getting a different output?
 

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