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juantheron
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$\displaystyle (1)\;\; \int_{-\infty}^{\infty}\frac{x^2}{1+4x+3x^2-4x^3-2x^4+2x^5+x^6}dx$
Integral from negative infinity to infinity
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SUMMARY
The integral from negative infinity to infinity of the function $\frac{x^2}{1+4x+3x^2-4x^3-2x^4+2x^5+x^6}$ evaluates to $\pi$. This result is derived using techniques from complex analysis, specifically residue theory, which allows for the evaluation of improper integrals. The discussion highlights the significance of understanding the behavior of the polynomial in the denominator to determine the poles and their contributions to the integral.
PREREQUISITES- Complex analysis, particularly residue theory
- Understanding of improper integrals
- Familiarity with polynomial functions and their roots
- Knowledge of contour integration techniques
- Study residue theory in complex analysis
- Learn about contour integration methods
- Explore the evaluation of improper integrals
- Investigate the properties of polynomials and their roots
Mathematicians, students of advanced calculus, and anyone interested in complex analysis and integral evaluation techniques.
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