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Math Help - Stuck on a tricky integral

  1. #1
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    Stuck on a tricky integral

    I tried get a tidy image of the integral with Integrator, but it told me the answer does not exist. I'm almost certain it does however.

    (1+ln(x))*sqrt(1+(xln(x))^2)

    Help! I tried integration by parts but it turns into a jumbled mess, unless I'm overlooking something.
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  2. #2
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    Wait a second, I just had some inspiration. I'll let you know If I truley need help...
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  3. #3
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    Ok, hmmm. I got it down to the integral of: sqrt(1+u^2) by setting xlnx=u and 1+lnx dx = du. Im stuck now How do I integrate sqrt(1+u^2) ?
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  4. #4
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    Quote Originally Posted by phack View Post
    Ok, hmmm. I got it down to the integral of: sqrt(1+u^2) by setting xlnx=u and 1+lnx dx = du. Im stuck now How do I integrate sqrt(1+u^2) ?
    Good job spotting the substitution. Now what you want to do is let u = sinh(y). Then du = cosh(y)dy, so your integral turns into:
    \int (1 + xln(x)) \sqrt{1 + (x ln(x))^2)} dx = \int \sqrt{1 + u^2} \, du = \int \sqrt{1 + sinh^2(y)} \cdot cosh(y)dy = \int cosh^2(y) dy

     = \int \frac{1}{4}(e^{2y} + e^{-2y} + 2)dy = ...

    You can finish this from here.

    -Dan
    Last edited by MathGuru; February 10th 2007 at 03:16 PM.
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