Need help with this definite integral

Click For Summary
SUMMARY

The integral $$\int_{0}^\infty \frac{x^2 \, dx}{x^4+(a^2+\frac{1}{b^2})x^2+\frac{2a^2}{b^2}}$$ can be approached using the residue theorem, although the roots of the denominator are complex. The discussion suggests that partial fraction decomposition may be a more effective method, particularly since the denominator can be expressed in terms of \(x^2\). No singularities exist in the integration region, indicating that a suitable contour can be chosen for evaluation.

PREREQUISITES
  • Understanding of definite integrals and improper integrals
  • Familiarity with complex analysis and the residue theorem
  • Knowledge of partial fraction decomposition techniques
  • Basic algebraic manipulation of polynomials
NEXT STEPS
  • Study the application of the residue theorem in complex integrals
  • Learn about partial fraction decomposition for rational functions
  • Explore techniques for evaluating improper integrals
  • Investigate polynomial root-finding methods for complex coefficients
USEFUL FOR

Mathematicians, engineering students, and anyone involved in advanced calculus or complex analysis who seeks to solve integrals involving rational functions.

jazhemar
Messages
3
Reaction score
0
I'm having a tough time with this integral:

$$\int_{0}^\infty \frac{x^2 \, dx}{x^4+(a^2+\frac{1}{b^2})x^2+\frac{2a^2}{b^2}}$$
where $$a, b \in \Bbb R^+$$ I tried using the residue theorem, but the roots of the denominator I found are quite complicated, and I got stuck.

What contour should I use? Is there an alternative method? I would appreciate any advice.
 
Physics news on Phys.org
There are no singularities in the region of integration. This looks like partial fractions might be the proper approach.
 
For partial fractions, the denominator terms will be in terms of x^2\ not\ x.
 

Similar threads

High School Straightforward integration…
  • · Replies 27 ·
Replies
27
Views
3K
Undergrad Substitution in a definite integral
  • · Replies 6 ·
Replies
6
Views
3K
Graduate A question about a complex integral
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Definite integral is undefined and not undefined
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Express the limit as a definite integral
  • · Replies 16 ·
Replies
16
Views
4K
Graduate Challenging integral involving exponentials and logarithms
  • · Replies 14 ·
Replies
14
Views
3K
High School Why is this definite integral a single number?
  • · Replies 20 ·
Replies
20
Views
4K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Ambiguity of the term "indefinite integral"
  • · Replies 11 ·
Replies
11
Views
2K