What is the Integral of 1/(x^4+1) from 0 to b?

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Homework Help Overview

The problem involves finding the integral of the function 1/(x^4 + 1) from 0 to b, with the upper limit approaching infinity. The subject area pertains to calculus, specifically integration techniques and improper integrals.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to begin the integration process. Some participants suggest using substitutions or decomposing the function into partial fractions. Others mention the use of computational tools like MATLAB to evaluate the integral.

Discussion Status

Participants are exploring various methods to approach the integral, with some hints provided regarding techniques such as partial fraction decomposition and the use of arctangent derivatives. There is an acknowledgment of the complexity of the problem, and multiple interpretations of the approach are being considered.

Contextual Notes

The original poster notes a lack of familiarity with certain techniques that may be necessary for solving the integral, indicating a potential gap in their learning. There is also a reference to the problem being tedious, suggesting that it may require significant effort to work through.

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Homework Statement


i do not know where to start with this one can anyone point me where to go please?

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

lower limit = 0 upper = infinity


Homework Equations



like i said i don't know where to go with this one.
i mean i know I am going to have to do integrate from 0 to b but i don't know where to start here.

The Attempt at a Solution

 
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It seems really messy if you integrate using substitutions ... etc.

I would recommend using f = 1/(x^2.y^2+1)

some MATLAB mess

>> f = 1/(x^4+1)

f =

1/(x^4+1)


>> int(f,x,0,inf)

ans =

1/4*pi*2^(1/2)


>> int(f,x)

ans =

1/8*2^(1/2)*log((x^2+x*2^(1/2)+1)/(x^2-x*2^(1/2)+1))+1/4*2^(1/2)*atan(x*2^(1/2)+1)+1/4*2^(1/2)*atan(x*2^(1/2)-1)
 
o wait i think i need to compare i think i have it ill repost. thanks rootX seeing your answer made me see that we have yet to really learn that, so it can't be. thanks again. ill post with what i get to see if I am right.
 
Here some hints to get you going. You'll have to decompose it to partial fractions, complete a square, and then use the derivative for arctan f(x). It's quite tedious, but most certainly doable.
 


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cont.
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