An improper integral is an integral where one or both of the limits of integration are infinite or where the integrand function is not defined at one or more points within the interval of integration.
To determine the convergence or divergence of an improper integral, you need to evaluate it using a limit. If the limit exists and is a finite number, then the integral converges. If the limit does not exist or is infinite, then the integral diverges.
Yes, it is possible for an improper integral to have a finite value even if it diverges. This can happen if the integral has a removable discontinuity or if the integral has a jump discontinuity at one or both of the limits of integration.
The comparison test is used to determine the convergence or divergence of an improper integral by comparing it to another integral whose convergence or divergence is already known. If the known integral converges and the given integral is greater than or equal to it, then the given integral also converges. If the known integral diverges and the given integral is less than or equal to it, then the given integral also diverges.
Yes, there are several techniques for evaluating improper integrals, including the limit comparison test, the ratio test, and the Cauchy condensation test. Additionally, some integrals can be evaluated using integration by parts or substitution. It is important to carefully consider the limits of integration and the properties of the integrand when using these techniques.