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Math Help - Hyperbolic functions

  1. #1
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    Hyperbolic functions

    I have to find the integral of 1/((25x^2-49)^1/2) using hyperbolic functions. I'm not sure how to start to know which functions to use, thanks
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  2. #2
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    It sounds like you are expected to know (or be able to determine from a table) the derivatives of the inverse hyperbolic functions. In particular, the one that is useful in this problem would be:

    \frac{d}{dx} \cosh^{-1}(x) = \frac{1}{\sqrt{x^2-1}}

    (for x>1)


    Can you solve this with this information?
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  3. #3
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    I wrote x as 7cosh(u) and ended up with arccosh(x/2) between 7 and 14, i'm not sure that this is right?
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  4. #4
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    You didn't say what the original endpoints were.

    I'm not sure which simplification method you use, but this is how I get it:

    \int \frac{1}{\sqrt{25x^2-49}} dx

    =\frac{1}{7} \int \frac{1}{\sqrt{\tfrac{25}{49}x^2-1}} dx

    =\frac{1}{7} \int \frac{1}{\sqrt{\left(\tfrac{5}{7}x\right)^2-1}} dx

    Let u=\tfrac{5}{7}x and du=\tfrac{5}{7}dx.

    =\frac{1}{5} \int \frac{1}{\sqrt{u^2-1}} du

    =\tfrac{1}{5} \cosh^{-1} u + C

    =\tfrac{1}{5} \cosh^{-1}\left(\tfrac{5}{7}x\right) + C


    And this is valid for u>1 \implies x>\tfrac{7}{5}.
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