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Math Help - Integration by parts

  1. #1
    Senior Member mollymcf2009's Avatar
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    Integration by parts

    Hi everyone!

    I am having trouble doing this integration by parts. Mine got really ugly and it doesn't seem like is should. Can y'all work this for me?

    \int\limits_{1}^{8} 5\cdot\frac{(\ln{x})^2}{x^3} dx

    Thanks!
    Molly
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Yo, Molly.

    Read my signature about how to increase your speed when integrating by parts.

    Now, if you want to make it to the classic way, put u=\ln^2x and dv=\frac{dx}{x^3}. (You're gonna have to make another integration by parts after this one.)
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  3. #3
    Senior Member mollymcf2009's Avatar
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    WOW!! Thanks so much for that!
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  4. #4
    Senior Member mollymcf2009's Avatar
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    Using that method you showed me, do I have this right?

    \int\limits_{1}^{8}  5 \cdot (lnx)^2 \cdot (x^{-3})' dx

    \frac{5}{2} \cdot (\frac{(lnx)^2}{x^2} + \frac{2lnx}{2x^3}) evaluated from 1 to 8
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  5. #5
    MHF Contributor
    skeeter's Avatar
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    let t = \ln{x}

    x = e^t

     <br />
dt = \frac{1}{x} \, dx<br />

     <br />
\int \frac{(\ln{x})^2}{x^2} \cdot \frac{1}{x} \, dx<br />

    substitute ...

     <br />
\int \frac{t^2}{e^{2t}} \, dt<br />

     <br />
\int t^2 \cdot e^{-2t} \, dt<br />

    now use parts ... in fact, tabular integration works well.
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