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Integral of 1/(1+x^4) dx

  1. Mar 21, 2013 #1
    ∫1/(1+x^4) dx, from 0 to ∞

    I have tried integration per partes, several different substitutions and transformation into different coordinate system but i have always only found another equivalent integral that i was not able to solve... I have also performed a numerical integration, but i need an analytical solution...
  2. jcsd
  3. Mar 21, 2013 #2


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    You should be able to work out the indefinite integral with one substitution. By the way, what is the derivative of the arc tangent?
  4. Mar 21, 2013 #3

    Ray Vickson

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    Partial fractions.
  5. Mar 22, 2013 #4


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    Hint: ##x^4 + 1 = (x^2+1)^2 - (\sqrt{2}x)^2##. Rearrange, factorise.

    Then use what Ray suggested.
  6. Mar 23, 2013 #5
    Try substituting ##\sqrt{u}=x##.

    Edit: Sorry, this doesn't actually work. Disregard!
    Last edited: Mar 23, 2013
  7. Mar 23, 2013 #6


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    You have a definite integral. I'd suggest using contour integration and the residue theorem if you know that technique.
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