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

  • Thread starter ritwik06
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  • #1
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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.
 

Answers and Replies

  • #2
580
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Is it easy to find the integral of (x^2)/(1+x^4)
Please help me???
 
  • #3
64
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I have not tried this, but you could try factoring 1+x^4 into two quadratics, then do partial fraction decomposition.
 
  • #4
580
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[tex]\int\frac{x^{2}}{1+x^{4}}[/tex]

Can this be determined????
 
  • #5
580
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I have not tried this, but you could try factoring 1+x^4 into two quadratics, then do partial fraction decomposition.
its difficult to factorize [tex]x^{4}+1[/tex].
Moreover,
can u please tell me about partial fractions decomposition? I havent learned it yet.
 
  • #6
HallsofIvy
Science Advisor
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[tex]\int\frac{x^{2}}{1+x^{4}}[/tex]

Can this be determined????
Have you tried the obvious substitution u= x4+ 1??
 
  • #7
580
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Have you tried the obvious substitution u= x4+ 1??
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:
  • #8
580
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Please help? If u think it cant be done with substitution. Please tell me about the other method, if any. Just a brief idea.
 
  • #9
64
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[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.
 
  • #10
580
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[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.
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!
 
  • #11
580
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Anyone gonna help me out? Please!
 
  • #12
64
0
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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