Final answer:
The question involves evaluating an improper integral of a rational function, which requires the identification of vertical asymptotes and the careful evaluation of limits to determine if the integral converges or diverges.
Step-by-step explanation:
The question is based on evaluating the integral ∫⁻¹² 2x + 1 / ∛ x² + x -6 dx, which involves the concept of improper integrals. Improper integrals occur when either the interval of integration is unbounded, or when the integrand is not defined at some point within the interval of integration, such as when a vertical asymptote is present. In this case, the integrand has a vertical asymptote where the denominator equals zero, which can be found by solving the equation x² + x - 6 = 0. Once the points of discontinuity are identified, the integral can be split at these points and evaluated as a limit.
Improper integrals such as this one require careful evaluation of limits to determine if they converge (result in a finite number) or diverge (go off to infinity). If an odd function is being integrated over a symmetric interval about the y-axis, the integral will equal zero due to the symmetric properties of odd functions since the positive area above the x-axis will cancel out the negative area below it.
However, in this example, the presence of a rational function with a cubic root makes the evaluation more complex and requires the limits of integration be observed closely to ensure accurate results. When dealing with improper integrals, it is important to analyze the behavior of the function at the points of discontinuity and approach the evaluation with valid reasoning to decide on convergence or divergence.