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

  • Thread starter transgalactic
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In summary: So if you 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, then you can use that information to simplify the second integral.
  • #1
transgalactic
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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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  • #2
As far as I can tell, after some cheating, you're supposed to somehow figure out that [tex] x^4 + 1 = (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1) [/tex]. I'm not sure exactly how one would easily recognize that though (assuming I didn't make a mistake).
 
  • #3
Well let's first put it in LaTeX for the benefit of the homework helpers:

[tex]\int \frac{x^2 +1}{x^4 + 1} dx[/tex]

How exactly did you cheat Mystic?
 
  • #4
Wolfram's Integrator.
 
  • #5
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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  • #6
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.
 
  • #7
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.

[tex]x^4+2x^2+1-2x^2[/tex]

[tex](x^2+1)^2-(\sqrt{2}x)^2[/tex]

[tex](x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)[/tex]
 
  • #8
Wow, I feel silly now. Thanks.
 
  • #9
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 ...

[tex]\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[/tex]

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 ...

[tex]\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[/tex]

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 [tex]\frac{A}{(x^2+\sqrt{2}x+1)}+\frac{B}{(x^2-\sqrt{2}x+1)}[/tex]
is capable of giving the form [tex]\frac{(x^2 + 1)}{(x^4 + 1)} [/tex] 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 [tex] x^{4}+1 =(x^2 +i) (x^2 -i) [/tex], partial fraction decomposition is simply

[tex] \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 } [/tex] 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:

[tex]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][/tex]

And also that:

[tex]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][/tex]

You get two integrals involving arctan.
 

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

What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to solve problems involving finding the total value of a quantity that changes continuously over a range. In other words, it is a way to find the total accumulation of a quantity over a certain interval.

How do you solve an integral?

To solve an integral, you need to use integration techniques such as substitution, integration by parts, or partial fractions. First, you need to identify the function that you are integrating and then apply the appropriate integration technique. It is also important to understand the properties and rules of integration to solve the integral correctly.

What is the general formula for solving integrals?

The general formula for solving integrals is ∫f(x)dx = F(x) + C, where f(x) is the function being integrated, F(x) is the antiderivative of f(x), and C is the constant of integration. This formula is known as the Fundamental Theorem of Calculus and is used to find the indefinite integral of a function.

How do you solve a complex integral question?

To solve a complex integral question, you need to break it down into smaller, more manageable parts. This can be done by using integration techniques and applying them to each part of the integral. It is also important to simplify the question as much as possible and use algebraic manipulation to solve for the integral.

What is the specific approach to solving the integral question ((x^2)+1)/((x^4)+1)?

The specific approach to solving this integral question is to use partial fractions to separate the fraction into smaller, simpler fractions. Then, use substitution to solve for each fraction individually. Finally, combine the solutions to get the final answer for the integral. It is also important to check for any extraneous solutions and to use the limits of integration to determine the final answer.

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