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Homework Help: Stuck on an integral

  1. Mar 21, 2007 #1
    1. The problem statement, all variables and given/known data
    [tex] \int\frac{1}{t^4+16}\,dt [/tex]


    2. Relevant equations
    Various integration formulas. I am pretty stuck.


    3. The attempt at a solution

    I have tried various substitutions, with no success. I tried t^2 = 4u, hoping to get it into the form 1/(u^2+1) so I could use the arctan formula, but then du/dt = t/2 and so I don't have an equivalent expression when I substitute.

    I have tried various other substitutions with no better luck. If anyone has a suggestion, I would be so grateful. This is the only one from this set I haven't been able to solve.

    Thanks!
    Brett
     
  2. jcsd
  3. Mar 21, 2007 #2

    dextercioby

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    HINT: [tex] t^4 +16=t^4 +8t^2 +16 -8t^2 [/tex]
     
  4. Mar 21, 2007 #3
    Thanks dextercioby. If I understand your hint, I would factor [tex] (t^2+4)^2-8t^2 [/tex] into two factors, use partial fractions, and then integrate the two expressions that would result?
     
  5. Mar 21, 2007 #4
    That doesn't seem to work. I end up with imaginaries in the expression, unless I am doing something wrong.
     
  6. Mar 21, 2007 #5
    I don't want to put you off- but the Integrator gives an answer that is quite complicated- involving 2 logs and 2 inverse tan functions. Could it be that you've misread the question?

    I would rate this integral as tough.
     
  7. Mar 21, 2007 #6
    Ok, I think I've got it... Wow. This is so much more complicated than any other expression. Wow.
     
  8. Mar 21, 2007 #7

    Gib Z

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    I'm not sure if thats what dex wanted.

    But either way, you're doing something wrong.

    [tex]\frac{1}{t^4+16}= \frac{\frac{-t}{16\sqrt{2}} - \frac{1}{8}}{t^2 + 2\sqrt{2} +4} - \frac{\frac{1}{8} - \frac{t}{16\sqrt{2}}}{-t^2+2\sqrt{2}-4}[/tex]. Not very nice.
     
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