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Math Help - Difficult integration using hyperbolic trig functions

  1. #1
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    Difficult integration using hyperbolic trig functions

    I need to calculate the following: \int \sqrt{2\cosh t(\cosh t-1)} dt, assuming t\ge0. So far I have tried some things which, quite messily, arrive at a solution but I cannot simplify my result back to the solution in the book, so I think I have taken a more difficult approach.

    Here's what I've done:
    \int \sqrt{2\cosh t(\cosh t-1)} dt=\int 2 \sinh(\tfrac t 2) \sqrt{\cosh t}dt=\tfrac 1 {\sqrt{2}} \int(e^{t/2}-e^{-t/2}) \sqrt{e^{-t}+e^t}dt
    This can then be solved using a u-substitution, such as u=e^t, but the result is really ugly, and I don't think this is the path the book would have intended for me to arrive at the simpler result.

    I would appreciate a hint. I'm guessing that there's some hyperbolic identity I'm forgetting.
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  2. #2
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    Re: Difficult integration using hyperbolic trig functions

    I think I figured it out;

    \int 2 \sinh(\tfrac t 2) \sqrt{\cosh t}dt = \int 2 \sinh(\tfrac t 2) \sqrt {2 \cosh ^2 (\tfrac t 2) - 1} dt

    Now let u=\cosh (\tfrac t 2) and proceed...

    I'm not seeing where I can mark this thread as solved... if anyone wants to point that out to me I would happily do so.
    Last edited by process91; April 23rd 2012 at 08:11 AM.
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