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

[jnazor]

[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.

 

[cristo]

Well what do you know about the method of residues? You've not really shown any work now, have you?

 

[jnazor]

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.

 

[cristo]

Well done!