How can the method of residues be used to solve integrals?

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Homework Help Overview

The discussion revolves around evaluating the integral of 1/(x^4+x^2+1) from 0 to infinity using the method of residues. The subject area includes complex analysis and integral calculus.

Discussion Character

  • Exploratory, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to find the poles of the function by solving the equation x^4+x^2+1=0 but struggles with obtaining useful results. Some participants question the completeness of the original poster's work and suggest that more detail is needed.
  • Another participant shares their experience of finding the poles and evaluating the residues, noting a simplification by substituting y=x^2 to facilitate solving the quartic equation.

Discussion Status

The discussion has progressed with one participant successfully finding the poles and evaluating the residues, while others have engaged in clarifying the process and exploring different approaches. There is acknowledgment of the original poster's initial challenges, but also recognition of the eventual resolution by another participant.

Contextual Notes

There appears to be an emphasis on the method of residues and its application to complex integrals, with some participants reflecting on the assumptions made during the problem-solving process.

jnazor
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[SOLVED] Integral using residues

Homework Statement


Show integral of 1/(x^4+x^2+1) from 0 to infinity equals (pi*sqrt(3))/6 using method of residues



Homework Equations





The Attempt at a Solution


Tried to solve x^4+x^2+1=0. Don't get very useful numbers for poles.
 
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Well what do you know about the method of residues? You've not really shown any work now, have you?
 
don't worry i worked it out. I didn't solve the quartic properly to get the poles before. I got my 4 poles two of which I am parts are more than 0. I evaluated the residues at those two poles and got the value. Since it's even I can divide that number by two to get the integral from 0 to inf rather than -inf to inf. I didn't notice there was a much easier way to solve that quartic by letting y=x^2 and then using quadratic formula to find y then x would be just the sqrt of y.
 
Well done!
 

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