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Math Help - Hyperbolic substitution

  1. #1
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    Hyperbolic substitution

    Hi,
    I am trying some revision for my maths test and I have to use hyperbolic functions to evaluate this integral.

    <br />
\int {\frac{3x}{\sqrt{{x^4-9}}}dx }<br /> <br />

    I am having trouble with the substitution. I tried letting

    <br /> <br />
x^2 = 3cosh(u)<br /> <br /> <br />
    But I still cant manage to find a solution. I suspect something nice is spose to happen with the 3x on the top of the integral but not sure how. Any help would be greatly appreciated.

    Thanks
    Elbarto



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  2. #2
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    Quote Originally Posted by elbarto View Post
    Hi,
    I am trying some revision for my maths test and I have to use hyperbolic functions to evaluate this integral.

    \int {\frac{3x}{\sqrt{{x^4-9}}}dx }

    I am having trouble with the substitution. I tried letting
    x^2 = 3cosh(u)
    But I still cant manage to find a solution. I suspect something nice is spose to happen with the 3x on the top of the integral but not sure how. Any help would be greatly appreciated.

    Thanks
    Elbarto
    x^2 = 3 \cosh u \Rightarrow 2x = 3 \sinh u \, \frac{du}{dx} \Rightarrow dx = \frac{3 \sinh u}{2x} \, du.
    Last edited by mr fantastic; November 1st 2008 at 09:25 PM. Reason: Forgot the du
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    Thank you for the fast response but I am confused as to why you took the derivative of both sides of the expression. Is this the short hand method of taking the arcosh() of the function or something similar?
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  4. #4
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    Quote Originally Posted by elbarto View Post
    Thank you for the fast response but I am confused as to why you took the derivative of both sides of the expression. Is this the short hand method of taking the arcosh() of the function or something similar?
    I used implicit differentiation to get the derivative. The derivative is needed because when you do the substitution you have to substitute for dx as well.

    Note the edit to my first post. I forgot the du in the expression for dx.
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