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Math Help - Integration with u-sub and trig

  1. #1
    Junior Member symstar's Avatar
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    Integration with u-sub and trig

    \int \frac{x^4}{\sqrt{x^{10}-2}}dx

    Let: u=x^5 \text{  }du=5x^4dx

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

    Let: u=\sqrt{2}\sec\theta \text{  }du=\sqrt{2}\sec\theta\tan\theta d\theta

    =\tfrac{1}{5}\int\frac{\sqrt{2}\sec\theta\tan\thet  a}{\sqrt{2}\tan\theta}d\theta
    =\tfrac{1}{5}\int\sec\theta d\theta
    =\tfrac{1}{5}\ln{\left|\sec\theta+\tan\theta\right  |}+C
    =\tfrac{1}{5}\ln{\left|\tfrac{u}{\sqrt{2}}+\tfrac{  \sqrt{u^2-2}}{\sqrt{2}}\right|}+C
    =\tfrac{1}{5}\ln{\left|\tfrac{x^5}{\sqrt{2}}+\tfra  c{\sqrt{x^{10}-2}}{\sqrt{2}}\right|}+C

    The book claims that the answer is:
    =\tfrac{1}{5}\ln{\left|x^5+\sqrt{x^{10}-2}\right|}+C

    I could see where the discrepancy could come from, but I don't see the error in my work... what did I do wrong?
    Last edited by symstar; September 16th 2008 at 02:09 PM. Reason: Left out a sqrt
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  2. #2
    Moo
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    Hello !
    Quote Originally Posted by symstar View Post
    \int \frac{x^4}{\sqrt{x^{10}-2}}dx

    Let: u=x^5 \text{  }du=5x^4dx

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

    Let: u=\sqrt{2}\sec\theta \text{  }du=\sqrt{2}\sec\theta\tan\theta d\theta

    =\tfrac{1}{5}\int\frac{\sqrt{2}\sec\theta\tan\thet  a}{\sqrt{2}\tan\theta}d\theta
    =\tfrac{1}{5}\int\sec\theta d\theta
    =\tfrac{1}{5}\ln{\left|\sec\theta+\tan\theta\right  |}+C
    =\tfrac{1}{5}\ln{\left|\tfrac{u}{\sqrt{2}}+\tfrac{  \sqrt{u^2-2}}{\sqrt{2}}\right|}+C
    =\tfrac{1}{5}\ln{\left|\tfrac{x^5}{\sqrt{2}}+\tfra  c{\sqrt{x^{10}-2}}{\sqrt{2}}\right|}+C

    The book claims that the answer is:
    =\tfrac{1}{5}\ln{\left|x^5+\sqrt{x^{10}-2}\right|}+C

    I could see where the discrepancy could come from, but I don't see the error in my work... what did I do wrong?
    It is correct

    To get your book's answer :

    \tfrac{x^5}{\sqrt{2}}+\tfrac{\sqrt{x^{10}-2}}{\sqrt{2}}=\tfrac{1}{\sqrt{2}} \left(x^5+\sqrt{x^{10}-2}\right)

    Thus \tfrac{1}{5}\ln{\left|\tfrac{x^5}{\sqrt{2}}+\tfrac  {\sqrt{x^{10}-2}}{\sqrt{2}}\right|}+C=\tfrac 15 \ln\left|\tfrac{1}{\sqrt{2}} \left(x^5+\sqrt{x^{10}-2}\right)\right|+C

    Use the rule \ln(ab)=\ln(a)+\ln(b) :

    =\tfrac 15 \left(-\ln(\sqrt{2})+\ln\left|x^5+\sqrt{x^{10}-2}\right)\right|+C

    =\tfrac 15 \ln\left|x^5+\sqrt{x^{10}-2}\right|\underbrace{-\tfrac{\ln(\sqrt{2})}{5}+C}_{\text{this is a constant}}

    =\tfrac 15 \ln\left|x^5+\sqrt{x^{10}-2}\right|+C'

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  3. #3
    Junior Member symstar's Avatar
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    Ah, I see. Thanks!
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