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Integrate (x^3 + x^2)/(1 + x^4) using substitution?

  1. May 25, 2008 #1
    Integrate the following:

    (x^3+x^2)/(1+x^4)

    I have been taught only integration by substitution. My teacher told me that this can be solved using that ith some trick.

    I have tried for a long time. All that I can do was to convert the numerator to x^2(x+1)
    and the denominator to (x^2-1)^2+2x^2
    but without sucess.

    Next I tried to break the given thing into two terms. it helped a little but I was again stuck on the second term, ie. , i couldnt again find out the integration of x^2/(1+x^4)



    Please hlp me. Its maddening.
     
  2. jcsd
  3. May 25, 2008 #2
    Is it easy to find the integral of (x^2)/(1+x^4)
    Please help me???
     
  4. May 25, 2008 #3
    I have not tried this, but you could try factoring 1+x^4 into two quadratics, then do partial fraction decomposition.
     
  5. May 25, 2008 #4
    [tex]\int\frac{x^{2}}{1+x^{4}}[/tex]

    Can this be determined????
     
  6. May 25, 2008 #5
    its difficult to factorize [tex]x^{4}+1[/tex].
    Moreover,
    can u please tell me about partial fractions decomposition? I havent learned it yet.
     
  7. May 25, 2008 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Have you tried the obvious substitution u= x4+ 1??
     
  8. May 25, 2008 #7
    yes,
    it yields the following:
    0.25[tex]\int\frac{1}{u \sqrt[4]{(u-1)}}[/tex]du
    But does this help or more complicate the problem?
     
    Last edited: May 25, 2008
  9. May 25, 2008 #8
    Please help? If u think it cant be done with substitution. Please tell me about the other method, if any. Just a brief idea.
     
  10. May 25, 2008 #9
    [tex]1+x^4 = 1+2x^2+x^4 - 2x^2=(1+x^2)^2-2x^2=(1-\sqrt 2 x + x^2)(1+\sqrt 2 x+x^2)[/tex].

    As to partial fraction decomposition, I suggest you have a look in your textbook or at wikipedia or other webpages.
     
  11. May 25, 2008 #10
    Yeah, I have read now about partial fractions. It means to cahnge the given thing into sum of 2 fractions. But hile trying to change:
    [tex]\frac{x^{2}}{1+x^{4}}[/tex]
    in the form of
    [tex]\frac{A}{1-x\sqrt{2}+x^{2}}[/tex]+[tex]\frac{B}{1+x\sqrt{2}+x^{2}}[/tex]
    I get the following equation:
    [tex]x^{2}[/tex]=(A+B)x[tex]^{2}[/tex]+(A+B)(2[tex]\sqrt{x}[/tex]+1)

    I cannot solve further for A and B. Can u do it?

    Its one hell out of the other. What should I do? Please help me!!! and try it out before answering. Thanks for all efforts!
     
  12. May 25, 2008 #11
    Anyone gonna help me out? Please!
     
  13. May 25, 2008 #12
    Your fractions should be of the form:
    [tex]\frac{Ax+B}{1-x\sqrt{2}+x^{2}} + \frac{Cx+D}{1+x\sqrt{2}+x^{2}}[/tex].
     
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