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Math Help - Integrate cosh(x)^2

  1. #1
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    Integrate cosh(x)^2

    Good afternoon,

    I am working on my last surface area problem, and am kind of hung up on the following:

     <br />
\int{cosh(x)^2} dx<br />

    Wait... could that be re-written as?:

    \int{1+sinh(x)^2} dx

    That doesn't really help either...
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  2. #2
    Ted
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    Is it cosh(x^2) ?
    If so, this integral is unelementary.

    Do you mean cosh^2(x) ?
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  3. #3
    MHF Contributor chisigma's Avatar
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    The easiest way is the following identity...

    \displaystyle \cosh^{2} x = \frac{1}{2} + \frac{1}{2}\ \cosh 2x (1)

    Kind regards

    \chi \sigma
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  4. #4
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    It is in fact cosh^2(x) which I have written as cosh(x)^2

    Where is the identity derived from?
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  5. #5
    Ted
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    Its a well-known identity.
    You can prove it by using the definitions of the Heperbolic Functions.


    ******Edit:******

    Another way to do it:



    cosh^2(x) = \left( \dfrac{e^{x}+e^{-x}}{2} \right)^2=\dfrac{e^{2x}+e^{-2x}+2}{2}=\frac{1}{2}e^{2x}+\frac{1}{2}e^{-2x}+1



    and it is easy to integrate.
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  6. #6
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    Would that be the double angle formula for cos^{2}x=\frac{1+cos2x}{2}

    Pardon my ignorance, but are all of the trig identities applicable to the hyperbolic functions?
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  7. #7
    Ted
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    No. Not all of them.
    Also, sometimes there is a small difference like the signs.

    for example:

    cos(x+y)=cos(x)cos(y) - sin(x)sin(y)

    but

    cosh(x+y)=cosh(x)cosh(y) + sinh(x) sinh(y)
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  8. #8
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    Just to check my work...

    What is the surface area of f(x)=\frac{e^{x}+e^{-x}}{2} for 0 \leq x \leq 2 when revolved about the x-axis?

    I get:

    S=\pi(2+sinh(4))


    Is this a reasonable looking answer?
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  9. #9
    Senior Member roninpro's Avatar
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    Quote Originally Posted by MechEng View Post
    Would that be the double angle formula for cos^{2}x=\frac{1+cos2x}{2}

    Pardon my ignorance, but are all of the trig identities applicable to the hyperbolic functions?
    Actually, yes, with some slight modifications. Have a look at Hyperbolic function - Wikipedia, the free encyclopedia, and look for "Osborn's Rule".
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