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
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![**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.
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