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Integration problem

  1. Nov 12, 2008 #1
    1. The problem statement, all variables and given/known data
    find the indefinate integral of 1+x2/(1+x4)




    2. Relevant equations



    3. The attempt at a solution
    im having trouble trying to figure what to use to substitute into this equation
     
  2. jcsd
  3. Nov 12, 2008 #2

    Dick

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    I would do it by partial fractions. You can factor (x^4+1) into two real quadratics. The coefficients may not all be rational, but that's ok. Can you do that?
     
  4. Nov 12, 2008 #3

    Mark44

    Staff: Mentor

    I'd like to see that factorization. I've been under the impression for a long time that x^4 + 1 was irreducible over the reals.

    If we allow complex numbers, the fourth roots of -1 (solutions of x^4 = -1) are
    cos(pi/4) + i sin(pi/4)
    cos(3pi/4) + i sin(3pi/4)
    cos(5pi/4) + i sin(5pi/4)
    cos(7pi/4) + i sin(7pi/4)
    none of which are real.
     
  5. Nov 12, 2008 #4
    (1+x^2)/(1+x^4) = 1/(1+(sqrt(2)*x+1)^2) + 1/(1+(sqrt(2)*x-1)^2)

    I found this by tracking back from the answer, which is obtained using Maple.

    For how to get this relation without refering to the answer, I'm not sure.
     
  6. Nov 12, 2008 #5

    Dick

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    Ok. x^4+1=(x^2+sqrt(2)x+1)(x^2-sqrt(2)x+1). x^4+1 is irreducible over the RATIONALS, not over the REALS. Your roots group into two pairs of complex conjugates (of course). Use them to construct the quadratics.
     
    Last edited: Nov 12, 2008
  7. Nov 12, 2008 #6

    Mark44

    Staff: Mentor

    OK, let me rephrase: x^4 + 1 can't be factored into linear factors with real coefficients.
     
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