SUMMARY
The integral of 1/(x^4+x^2+1) from 0 to infinity equals (π√3)/6, as demonstrated using the method of residues. The solution involved identifying the poles of the quartic equation x^4+x^2+1=0, which were found by substituting y=x^2 and applying the quadratic formula. The residues at the relevant poles were evaluated, and due to the even nature of the function, the result was halved to obtain the desired integral from 0 to infinity.
PREREQUISITES
- Understanding of complex analysis and contour integration
- Familiarity with the method of residues
- Knowledge of polynomial equations and their roots
- Proficiency in evaluating integrals involving complex functions
NEXT STEPS
- Study the method of residues in detail, focusing on complex contour integration techniques
- Learn how to factor quartic polynomials and find their roots
- Explore the application of the quadratic formula in solving polynomial equations
- Practice evaluating integrals using the residue theorem with various complex functions
USEFUL FOR
Students and professionals in mathematics, particularly those studying complex analysis, as well as anyone interested in advanced techniques for solving integrals.