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Math Help - Integral

  1. #1
    MHF Contributor Mathstud28's Avatar
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    Integral

    I was doing some practice excersises for some diff eqs and this integral came up...now I got it right, but it was messy, can someone here give a better solution

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

    Now since the domain is every x greater than zero I rewrote this as

    \int\frac{\sqrt{1+x^2}}{x^2}dx

    Then said, Let x=\tan(\theta)

    which is not fun, not hard, but not fun.

    Is there a cool not obvious (or obvious) trick for this integral?
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by Mathstud28 View Post
    I was doing some practice excersises for some diff eqs and this integral came up...now I got it right, but it was messy, can someone here give a better solution

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

    Now since the domain is every x greater than zero I rewrote this as

    \int\frac{\sqrt{1+x^2}}{x^2}dx
    Let x=\sinh t \Longrightarrow \mathrm{d}x=\cosh t \,\mathrm{d}t

    <br />
\int\frac{\sqrt{1+x^2}}{x^2}\,\mathrm{d}x=\int\fra  c{\sqrt{1+\sinh^2t}}{\sinh^2t}\cosh t\,\mathrm{d}t=\int\frac{\cosh^2 t}{\sinh^2 t}\,\mathrm{d}t=\int\frac{1}{\sinh^2 t}+1\,\mathrm{d}t

    hence

    \int\frac{\sqrt{1+x^2}}{x^2}\,\mathrm{d}x=-\frac{1}{\tanh t}+t+C=-\frac{\sqrt{1+x^2}}{x}+\mathrm{asinh} x+C'

    I don't know if it's funnier than your method.
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  3. #3
    Math Engineering Student
    Krizalid's Avatar
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    Try a reciprocal substitution.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Krizalid View Post
    Try a reciprocal substitution.
    I am assuming you mean start from here

    \int\frac{\sqrt{1+x^2}}{x^2}dx

    Because that is what most look like when you suggest this method, alright well I will give it a go

    Let x=\frac{1}{u}

    So dx=\frac{-1}{u^2}

    So we have

    -\int\frac{\frac{1}{u^2}\sqrt{1+\frac{1}{u^2}}}{\fr  ac{1}{u^2}}=-\int\sqrt{1+\frac{1}{u^2}}du

    Is this the right start?
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  5. #5
    Math Engineering Student
    Krizalid's Avatar
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    Yes, now do the remaining algebra and you'll get an easy integral.
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