Difficult integration using hyperbolic trig functions

**process91** · April 22nd 2012, 06:45 PM

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.

**process91** · April 23rd 2012, 07:09 AM

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.

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