Question

Find $\int_{-\infty}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x$.

   Find $\int_{-\infty}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x$.

Finite Mathematics and Calculus with Applications

Finite Mathematics and Calculus with Applications

Margaret L. Lial,… 9th Edition

Chapter 16, Problem 36

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Solved by Expert Linh Vu

06/18/2020

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An odd function is a function that satisfies the property $f(-x) = -f(x)$ for all $x$ in the function's domain. Show more…

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Find $\int_{-\infty}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x$.

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Key Concepts

Symmetry in Integration

Symmetry in integration involves recognizing that the integrand possesses properties (such as being even or odd) that simplify the evaluation of definite integrals over symmetric intervals. By identifying symmetry, one can often reduce the computational effort needed or directly conclude the value of the integral, particularly in cases involving infinite symmetric limits.

Odd Function

An odd function is defined by the property f(?x) = ?f(x) for every x in its domain. When integrating an odd function over an interval symmetric about zero, the contributions from the negative and positive parts cancel each other out, resulting in an integral of zero. This property is especially useful for quickly evaluating integrals over symmetric limits without performing the complete integration process.

Improper Integrals

Improper integrals involve limits of integration that extend to infinity or integrals of functions that have unbounded behavior within the interval of integration. Evaluating these integrals often requires the use of limits to define the integral properly, as the infinite boundaries need to be approached in a limit sense to ensure that the integral converges to a finite value.

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