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Math Help - A formula for the integral of exponentials and hyperbolic trig functions

  1. #1
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    A formula for the integral of exponentials and hyperbolic trig functions

    Hello,
    Do we have any formula for
    Integral of \int{e^(ax)\cosh(x)dx}
    and also \int{e^(ax)\sinh(x)dx}
    Last edited by Hamed; March 7th 2011 at 06:51 AM.
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  2. #2
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    Quote Originally Posted by Hamed View Post
    Do we have any formula for
    Integral of e^(at)cosh(bx)dx?
    and also e^(at)sinh(bx)dx
    First, are you sure you have written what you meant?

    Then because \dfrac{d(\sinh(bx))}{dx}=b\cosh(bx) I don't see much to do.
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  3. #3
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    Assuming that you meant \displaystyle \int{e^{ax}\cosh{bx}\,dx} and \displaystyle \int{e^{ax}\sinh{bx}\,dx}, for each you need to use Integration by Parts twice.
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    Quote Originally Posted by Prove It View Post
    Assuming that you meant \displaystyle \int{e^{ax}\cosh{bx}\,dx} and \displaystyle \int{e^{ax}\sinh{bx}\,dx}, for each you need to use Integration by Parts twice.
    Please note in the OP it is \displaystyle \int{e^{at}\cosh{bx}\,dx} and [tex] not \displaystyle \int{e^{ax}\cosh{bx}\,dx}.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Please note in the OP it is \displaystyle \int{e^{at}\cosh{bx}\,dx} and [tex] not \displaystyle \int{e^{ax}\cosh{bx}\,dx}.
    Which is why I wrote "Assuming that you meant..."
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    Yes, but as you said before, that is very simple. Assuming it is in fact, e^{ax}sinh(bx), perhaps the simplest thing to do is to write it as
    e^{ax}\left(\frac{e^{bx}- e^{-bx}}{2}\right)= \frac{e^{(a+b)x}- e^{(a- b)x}}{2}
    and integrate that.
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    x not t
    for e^(ax)cos(bx)dx= [(e^(ax))/(a^2+b^2)]*(acosbx+bsinbx)

    Can we have sth for e^(ax)cosh(bx)dx?
    and e^(ax)sinh(bx)dx

    How can I use TEX?
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    Quote Originally Posted by Hamed View Post
    How can I use TEX?
    See here for a LaTeX tutorial.
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    Quote Originally Posted by Hamed View Post
    x not t
    for e^(ax)cos(bx)dx= [(e^(ax))/(a^2+b^2)]*(acosbx+bsinbx)
    Can we have sth for e^(ax)cosh(bx)dx?
    and e^(ax)sinh(bx)dx
    How can I use TEX?
    In that case use post #6. It is the best way to do it.
    There is a LaTeX tutorial here.
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    I need sth like that formula!
    Quote Originally Posted by Plato View Post
    In that case use post #6. It is the best way to do it.
    There is a LaTeX tutorial here.
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  11. #11
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    Quote Originally Posted by Hamed View Post
    I need sth like that formula!
    You should be able to do this for yourself.
    \displaystyle\int {\frac{1}<br />
{2}\left( {e^{\left( {a + b} \right)x}  - e^{\left( {a - b} \right)x} } \right)dx}  = \frac{1}<br />
{2}\left( {\frac{{e^{\left( {a + b} \right)x} }}<br />
{{a + b}} - \frac{{e^{\left( {a - b} \right)x} }}<br />
{{a - b}}} \right) + C
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  12. #12
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    Quote Originally Posted by Hamed View Post
    I need sth like that formula!
    I don't know what you mean by "sth', but if you need to write your indefinite integral in terms of sines and cosines, use integration by parts twice.

    Otherwise, if you want a quick way to perform the integration, first convert the hyperbolic functions to their exponential forms.
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