Convergence of Improper Integral from 0 to Infinity

Determine whether the integral from 0 to infinity of $\displaystyle \int_{0}^{\infty} \frac{x}{(x^2+2)^2} \, dx$ is convergent or divergent and evaluate it if it is convergent.

This problem involves evaluating the convergence of an improper integral, a type of integral that appears frequently in calculus and analysis. Improper integrals are integrals that involve infinite limits of integration or integrals of functions with discontinuities over some interval. The primary challenge in solving this problem is to determine whether the integral converges, and if it does, to calculate its value.

When addressing improper integrals, one useful strategy is to first consider the behavior of the integrand as it approaches the problematic points—in this case, as the variable approaches infinity. A typical approach is to analyze the integrand and see if it resembles any basic forms of known convergent or divergent integrals, such as a p-integral. Additionally, comparison tests can be employed to compare the given integral to another integral whose convergence properties are already known.

In this specific problem, the integrand is a rational function, which may suggest the use of partial fraction decomposition. However, due to the presence of the quadratic in the denominator raised to a power and the range of integration being from zero to infinity, the problem may be handled by considering the behavior at the boundaries using appropriate limit processes. This problem effectively combines skills from several areas of integration techniques and tests for convergence, making it an interesting exercise in analyzing and integrating rational functions over an infinite range.

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