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Math Help - Integral!

  1. #1
    Newbie AgentSmith's Avatar
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    Integral!

    \int \frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}dx

    please help...
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  2. #2
    MHF Contributor
    skeeter's Avatar
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    Re: Integral!

    Quote Originally Posted by AgentSmith View Post
    \int \frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}dx

    please help...
    Wolfram Alpha's attempt ...

    http://www.wolframalpha.com/input/?_...1&incTime=true
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  3. #3
    Member sbhatnagar's Avatar
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    Re: Integral!

    Let I= \int \frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}dx

    1. Note that:

    \begin{align*}I &= \int \frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}dx \\ &=  \int \frac{1-\frac{1}{x^2}}{\Big( x+\frac{1}{x}\Big)\sqrt{\left( x+\frac{1}{x}\right)^2-2}}dx\end{align*}

    2. Substitute u=x+\frac{1}{x} and du=1-\frac{1}{x^2}:

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

    3. Substitute u=\sqrt{2}\sec{\theta} and du = \sqrt{2} \sec{\theta}\tan{\theta} \ dx

    \begin{align*} I &= \frac{1}{\sqrt{2}}\int \ d \theta \end{align*}
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