Solving Integral Question: ((x^2)+1)/((x^4)+1)

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

i tride to solve it in every way possible
1.by splitting it into two polinomials
2.by subtsituting x^4=t
3.by puttinh x=tan t

nothing works

how do i solve it
 
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As far as I can tell, after some cheating, you're supposed to somehow figure out that x^4 + 1 = (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1). I'm not sure exactly how one would easily recognize that though (assuming I didn't make a mistake).
 
Well let's first put it in LaTeX for the benefit of the homework helpers:

\int \frac{x^2 +1}{x^4 + 1} dx

How exactly did you cheat Mystic?
 
Wolfram's Integrator.
 
Mystic998 said:
I'm not sure exactly how one would easily recognize that though (assuming I didn't make a mistake).
Buy Gelfand's Algebra book and you'll be able to factor almost any polynomial :p
 
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It's somewhat depressing getting recommendations for high school algebra books while sitting in my crappy three-person TA office.

Anyway, out of curiosity, how would one proceed in getting that factorization of x^4 + 1. I've honestly got no clue, and that would probably be handy information.
 
Gelfand may be intended for HS students, but his books are hard. I just finished Calc 2 and I still struggle on a lot of his problems.

x^4+2x^2+1-2x^2

(x^2+1)^2-(\sqrt{2}x)^2

(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)
 
Wow, I feel silly now. Thanks.
 
Mystic998 said:
Wow, I feel silly now. Thanks.
Exactly how I felt before learning a few things in his book :-x
 
  • #10
After splitting it into 2, and using Mystic's advice ...

\int\frac{x^2}{(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)}dx+\int\frac{1}{(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)}dx

Partial Fraction?
 
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  • #11
so how do i solve it after this step
 
  • #12
transgalactic said:
so how do i solve it after this step
You have to use Partial Fractions, have you learned that method yet?
 
  • #13
yes i did but it looks very scary
will it work?
 
  • #14
transgalactic said:
yes i did but it looks very scary
will it work?
Yes and it's pretty much the only method I can think of. After that, you may want to mess around with completing the square, I did that in order to use arctan.
 
  • #15
rocophysics said:
After splitting it into 2, and using Mystic's advice ...

\int\frac{x^2}{(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)}dx+\int\frac{1}{(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)}dx

Partial Fraction?

Have you done it? Note I think it saves some work if you do not split it as rocophysics suggests but just notice that \frac{A}{(x^2+\sqrt{2}x+1)}+\frac{B}{(x^2-\sqrt{2}x+1)}
is capable of giving the form \frac{(x^2 + 1)}{(x^4 + 1)} and that part of it is quite simple to get.

Edit: I mean easy to see or find what A and B must be.
 
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  • #16
Since x^{4}+1 =(x^2 +i) (x^2 -i), partial fraction decomposition is simply

\frac{x^{2}+1}{x^{4}+1} =\frac{1+i}{2}\frac{1}{x^2 +i} +\frac{1-i}{2}\frac{1}{x^2 -i } and then the integration would be easy.
 
  • #17
The remark of epenguin is correct, you can llok for A and B to make the integrals more easy to solve. However if you notice the following:

x^2+1=\frac{1}{2}(2x^2+2) = \frac{1}{2}\left[\left(x+\frac{\sqrt{2}}{2}\right)^2+\left(\frac{\sqrt{2}}{2}\right)^2\right] + \frac{1}{2}\left[\left(x-\frac{\sqrt{2}}{2}\right)^2+\left(\frac{\sqrt{2}}{2}\right)^2\right]

And also that:

x^4+1 = \left[\left(x+\frac{\sqrt{2}}{2}\right)^2+\left(\frac{\sqrt{2}}{2}\right)^2\right] \cdot \left[\left(x-\frac{\sqrt{2}}{2}\right)^2+\left(\frac{\sqrt{2}}{2}\right)^2\right]

You get two integrals involving arctan.
 
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