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Integration Question - Please Help

  1. Feb 17, 2005 #1
    Given Integral (Between 0 and infinity) sin ax/sinh bx = (pi/2b).tanh[(api)/(2b)]. Calculate:

    Integral (Between 0 and infinity) cos ax/sinh bx dx.

    Any ideas? Cheers.
     
  2. jcsd
  3. Feb 18, 2005 #2
    Do u know leibnitz rule of differentiation under integral sign?

    -- AI
     
  4. Feb 18, 2005 #3

    saltydog

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    Jesus. That one neither and I spent some time on it too. Thanks.

    Nima, can you report the answer please. Try and use LaTeX (look at the on-line reference for it in the group). Here, I'll start it for you, just select "quote" and add on:

    [tex]\int_0^\infty \frac{\cos(ax)}{\sinh(bx)}=[/tex]
     
    Last edited: Feb 18, 2005
  5. Feb 18, 2005 #4

    dextercioby

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    If you differentiate wrt "a" under the integral sign,i'd like to know how one gets read o the nasty "x" that will appear in the numerator...:confused:

    Daniel.
     
  6. Feb 18, 2005 #5

    saltydog

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    made mistake with formula

    Nima, where you at? I aint' letting this one go. Just differentiate it throughout using Leibniz's rule (and no I didnt' know to do this until they told me). Granted, you have to take the derivative of infinity but just think of the upper limit as a constant 'c' and at the limit as c goes to infinity, it's derivative is still zero. Else, I'll write it up tomorrow.

    This is Leibnitz's rule. Just fill in the blanks.


    [tex]\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)}G(x,t)dt=G(x,\beta(x))\frac{d\beta}{dx}-G(x,\alpha(x))\frac{d\alpha}{dx}+\int_{\alpha(x)}^{\beta(x)}\frac{\partial G}{\partial x}dt[/tex]


    Salty

    Edit: I made a mistake with the formula and corrected it. Now I understand what Daniel meant about the 'x' in the numerator. Will need to regroup . . .

    Sorry if I caused problems for anyone.
     
    Last edited: Feb 19, 2005
  7. Feb 19, 2005 #6
    salty,
    F(a) = some integral with respect to 'x' containing parameter 'a'
    F'(a) = some integral with respect to 'x' containing parameter 'a'

    Can u see what i am pointing at?
    My mention of Leibnitz's rule of differentiation was just to indicate that u can differentiate under integral sign, as such u dont explicitly need that formula for this problem :)

    -- AI
     
  8. Feb 19, 2005 #7

    saltydog

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    Alright, I'm stuck. I used Leibnitz rule incorrectly above. Next time I'll double-check things before I say anything Nima. Sorry. Gonna work on it some more and check it with real data before I post anything.
     
  9. Feb 19, 2005 #8

    saltydog

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    Seems to me, after looking at it some more, that the cos integral diverges but I don't want to cause more confussion for Nima or anyone else. Would still like to find the antiderivative to prove explicitly that it does. Am I correct in this analysis?

    I mean, it would have been a more interesting question if we needed to find out what [itex]\frac{\cos(ax)}{\cosh(bx)}[/itex] was. That's just me though.
     
    Last edited: Feb 19, 2005
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