Convergence of Improper Integral from 1 to Infinity

Consider the integral from 1 up to infinity of rac{x-2}{x^3+1}. Determine if this integral converges or diverges.

When determining if an improper integral converges or diverges, such as the integral from 1 to infinity of rac{x-2}{x^3+1}, a strategic approach is crucial. An improper integral is characterized by an infinite interval, an unbounded region, or a discontinuous integrand. For these problems, understanding the behavior of the function as x approaches infinity can provide insights into convergence or divergence.

Comparison tests are often instrumental; comparing to a known integral can quickly suggest behavior. Alternatively, using limit processes to transform the problem into something tractable is another avenue. Here, the function's asymptotic behavior is considered: as x tends to infinity, examining the dominant terms in the numerator and denominator offers clarity about how the function behaves.

The denominator grows much faster than the numerator, which typically suggests convergence, similar to a p-series with p greater than 1. This integral is a gateway to broader applications of convergence tests and foundational within calculus for tackling infinite series problems.