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Thread: Antiderivative involving hyp. functions.

  1. #1
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    Antiderivative involving hyp. functions.

    I haven't much experience with hyperbolic functions, having never learned them, but the result of this integral seems to involve one:

    $\displaystyle \int{\sqrt{1+u^2} du}$

    Their solution is:

    $\displaystyle \frac{1}{2}u\sqrt{1+u^2}+\frac{1}{2}\text{arsinh}{ u}$

    (What's the proper latex command for arsinh?)

    I'm not sure how they derived that.
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  2. #2
    Super Member Showcase_22's Avatar
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    They've used the identity $\displaystyle cosh^2x-sinh^2x=1$.

    Remember that $\displaystyle \frac{d}{dx}cosh(x)=sinh(x)$ and $\displaystyle \frac{d}{dx}sinh(x)=cosh(x)$.
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  3. #3
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    Ok, so using that identity, would they then have done:

    $\displaystyle \int{\sqrt{u^2+\cosh^2(u)-\sinh^2(u)}}du$ ?

    If so, would this be integrated by parts?

    But I'm still not sure how they ended up with an Inverse Hyperbolic Sine (arsinh). Since the $\displaystyle \int{\sinh(x)}dx = \cosh(x)$, right? I don't see where arsinh comes into this.
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  4. #4
    Super Member Showcase_22's Avatar
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    Solve $\displaystyle
    \int{\sqrt{1+u^2} du}
    $.
    Let $\displaystyle u=sinh(x)$. Therefore $\displaystyle \frac{du}{dx}=cosh(x)$.

    The integral becomes: $\displaystyle \int\sqrt{1+sinh^2(x)} cosh(x) \ dx$

    $\displaystyle \int \sqrt{cosh^2(x)} cosh(x) \ dx$

    $\displaystyle \int cosh^2(x) \ dx$

    Now use the identity $\displaystyle cosh^2(x)+sinh^2(x)=cosh(2x)$ and $\displaystyle cosh^2(x)-sinh^2(x)=1$ to get $\displaystyle 2cosh^2(x)-1=cosh(2x)$.

    Since we require $\displaystyle cosh^2(x)$ we can rearrange this to get $\displaystyle cosh^2(x)=\frac{cosh(2x)+1}{2}$.

    We now need to integrate $\displaystyle \frac{1}{2}\int cosh(2x)+1dx=\frac{1}{2}(\frac{1}{2}sinh(2x)+x+C) $

    $\displaystyle sinh(2x)=2sinh(x)cosh(x)$ but using $\displaystyle cosh^2(x)-sinh^2(x)=1$ we get that $\displaystyle cosh(x)=\sqrt{1+sinh^2(x)}$. Combining this with the other identity gives $\displaystyle sinh(2x)=2sinh(x)\sqrt{1+sinh^2(x)}$.

    We also know that $\displaystyle x=arsinh(u)$ from how we defined u at the very start.

    Therefore $\displaystyle \frac{1}{2}(\frac{1}{2}sinh(2x)+x)+K$$\displaystyle =\frac{1}{2}(\frac{1}{2}2sinh(x)\sqrt{1+sinh^2(x)} +x)+K=\frac{1}{2}u\sqrt{1+u^2}+\frac{1}{2}arsinh(u )+K$.

    In this case, they've taken $\displaystyle K=C=0$.
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  5. #5
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    Thank you very much for the explanation
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