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I Griffiths problem 2.7 Messy Integral

  1. Sep 7, 2016 #1
    Hi all,


    twRFdMD.png

    "Integral can be done by partial fractions - or look it up" So second line, that's what I want to do.

    How to deal with this? What substitution can I use? Never encountered partial fractions with non-integer exponents.

    Someone give me a tip?

    Thanks in advance
     
  2. jcsd
  3. Sep 7, 2016 #2

    Charles Link

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    Homework Helper

    I do think the author must have meant integration by parts. The term without a ## u ## in the numerator that has a ## (A+Bu)^{3/2} ## in the denominator is a straightforward integration and the ## u \, du/(A+Bu)^{3/2} ## can be readily integrated by parts.
     
  4. Sep 7, 2016 #3
    You can directly use integration by parts.
     
  5. Sep 7, 2016 #4
    I get something extremely messy. Here is the first part, the second part with integration by parts,I'll do it again tomorrow.

    So the first term

    integral (z du / (R² + z² - 2 Rz u) ^3/2 )

    d(R² + z² - 2 Rz u) = -2 R z du

    The integral becomes

    -1/2R * integral ( d(R² + z² - 2 Rz u) / (R² + z² - 2 Rz u) ^3/2) )

    Then I get


    1/R * (1/ (R² + z² -2Rzu) ^ 1/2)

    The second part I'm going to do again tomorrow, and I'm going to try to put it in latex (have to learn latex quick) because posting like this is unreadable. But the first part is correct right?
     
  6. Sep 7, 2016 #5
    You can differentiate z - Ru and integrate ( du / (R² + z² - 2 Rz u)^3/2)
     
  7. Sep 7, 2016 #6

    mathman

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    Science Advisor
    Gold Member

    From first to second line is simply the substitution given. The integration is from second line to third.
     
  8. Sep 10, 2016 #7

    Ok, so partial integration. integral (fdg) = fg - integral (gdf)

    f = z-Ru
    dg = du / (R² +z² -2zRu) ^ 3/2

    For g I get
    1 / ( (Rz) * (R² + z² -2*R*z*u)^1/2)

    fg = (z-Ru) / ( (Rz) * (R² + z² -2*R*z*u)^1/2)

    now the integral part - integral ( g d f)

    df = -R du

    So

    The integral to solve

    Rdu / ((Rz) * (R² + z² -2*R*z*u)^1/2))

    Becomes

    1/z * (2/-2*R*z) * (R² + z² -2*R*z*u)^1/2

    Combining two terms, and rearranging

    I get solution


    DEAR GOD

    hhahaha

    (Okay I really need to learn latex now haha)


    Thank u also.

    How to give everyone upvotes in this thread?
     
  9. Sep 10, 2016 #8
    You can hit the like button in the bottom right corner of each post.
     
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