Using residues, evaluate the integral -8 x² (x² + 1)(x² +9) dx
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
Using residues, evaluate the integral
\[
\int_{-\infty}^{\infty} \frac{x^2}{(x^2 + 1)(x^2 + 9)} \, dx
\]
**Explanation:**
- This integral is to be evaluated over the entire real line.
- The function \( \frac{x^2}{(x^2 + 1)(x^2 + 9)} \) is a rational function that needs to be analyzed using complex analysis, specifically the method of residues.
- The poles of the integrand, which are the roots of the denominators \( x^2 + 1 = 0 \) and \( x^2 + 9 = 0 \), are critical in applying residue calculus. These poles are \( x = i, -i, 3i, -3i \) in the complex plane.
- The goal is to find the residues at these poles that lie in the upper half of the complex plane and use them to evaluate the original integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa62f7b35-7db6-46d8-92c3-a45ad2747ea7%2F90377651-2bc5-419b-ae16-fc620c1a2ab8%2F27u46q_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Using residues, evaluate the integral
\[
\int_{-\infty}^{\infty} \frac{x^2}{(x^2 + 1)(x^2 + 9)} \, dx
\]
**Explanation:**
- This integral is to be evaluated over the entire real line.
- The function \( \frac{x^2}{(x^2 + 1)(x^2 + 9)} \) is a rational function that needs to be analyzed using complex analysis, specifically the method of residues.
- The poles of the integrand, which are the roots of the denominators \( x^2 + 1 = 0 \) and \( x^2 + 9 = 0 \), are critical in applying residue calculus. These poles are \( x = i, -i, 3i, -3i \) in the complex plane.
- The goal is to find the residues at these poles that lie in the upper half of the complex plane and use them to evaluate the original integral.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
