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Math Help - hyperbolic integration

  1. #1
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    hyperbolic integration

    Just wondering if you could help me integrate the following:

    [(cosh (x^0.5)) / (x^0.5)] dx

    with respect to x

    cheers
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  2. #2
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    Hello, daaavo!

    \int \frac{\cosh\left(x^{\frac{1}{2}}\right)}{x^{\frac{  1}{2}}}\,dx

    We have: . \int\cosh\left(x^{\frac{1}{2}}\right)\cdot\frac{dx  }{x^{\frac{1}{2}}}

    Let u \:=\:x^{\frac{1}{2}} \quad\Rightarrow\quad du \:=\:\frac{1}{2}x^{-\frac{1}{2}}dx\quad\Rightarrow\quad\frac{dx}{x^{\f  rac{1}{2}}} \:=\:2\,du

    Substitute: . \int\cosh u\,(2\,du) \;\;=\;\;2\int\cosh u\,du \;\;=\;\;2\sinh u + C

    Back-substitute: . 2\sinh\left(x^{\frac{1}{2}}\right) + C

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  3. #3
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    You could rewrite \frac{cosh(\sqrt{x})}{\sqrt{x}}=\frac{\frac{1}{2}e  ^{\sqrt{x}}+\frac{1}{2}e^{-\sqrt{x}}}{\sqrt{x}}

    Now, let u=\sqrt{x}, \;\ 2du=\frac{1}{\sqrt{x}}dx

    You get:

    \int{e^{u}}du+\int{e^{-u}}du=2sinh(u)

    =\boxed{2sinh(\sqrt{x})}
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  4. #4
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    Thanks for your help - would you also possibly be able to guide me through this one too.... I'm trying to teach myself hyperbolic integration for fun, but the examples in my textbook are pretty poor...

    Integrate: 2x((x^2)+1)^23) dx

    Thanks very much for all of your help
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  5. #5
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    Quote Originally Posted by daaavo View Post
    Thanks for your help - would you also possibly be able to guide me through this one too.... I'm trying to teach myself hyperbolic integration for fun, but the examples in my textbook are pretty poor...

    Integrate: 2x((x^2)+1)^23) dx

    Thanks very much for all of your help
    No hyperbolic functions needed here.

    Make the substitution u = x^2 + 1. Then \frac{du}{dx} = 2x \Rightarrow dx = \frac{du}{2x}.

    Then the integral becomes:

    \int (2x) \, u^{23} \, \frac{du}{2x} = \int u^{23} \, du.

    So the answer will be \frac{1}{24} (x^2 + 1)^{24} + C.
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