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Math Help - What is this integral

  1. #1
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    What is this integral

    What is this integral
    \int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{  b}dx
    where a and b are constants.
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  2. #2
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    NOT Solved: What is this integral

    Quote Originally Posted by JulieK View Post
    What is this integral
    \int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{  b}dx
    where a and b are constants.
    Let Sin(X)=ax then dx=\frac{Cos(X)}{a}dX

    the integral becomes:

    \int\left(\frac{X}{Sin(X)}\right)^{b}\frac{Cos(X)}  {a}dX

    now

    \int\left({Sin(X)}\right)^{-b}{Cos(X)}dX= \frac {Sin(X)^{1-b}} {1-b}=u_1

    and the integral becomes

    \int \frac {X^bu_1'}{a}dX=\frac{u_1X}{a}-\int \frac{u_1bX^{b-1}}{a}dX=\frac{u_1X}{a}-\int \left( \frac{X}{Sin(X)}\right)^{b-1}} \frac {b}{a(1-b)}dX

    This looks like a dead end . But I include it because it might give you an idea?

    Perhaps there is no closed form solution? In my last line I am trying to integrate sinc(X) raised to the power of 1-b. But if you google 'integral of sinc' it seems like this is far from trivial. See here Sine Integral -- from Wolfram MathWorld for the integral of sinc.
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  3. #3
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    Re: What is this integral

    You have used the wrong substitution, as the OP's integrand was in terms of the inverse HYPERBOLIC sine function. So ax = sinh(t) is appropriate.
    Thanks from Kiwi_Dave
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