An improper integral with variable is an integral where one or both of the limits of integration are infinite, or where the integrand function has a vertical asymptote within the interval of integration. In these cases, the integral cannot be evaluated using traditional methods and requires special techniques to determine its value.
To determine if an improper integral with variable converges or diverges, you must evaluate the limit of the integral as the variable approaches the limits of integration. 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.
Some common techniques for evaluating improper integrals with variables include using the limit comparison test, the comparison test, the ratio test, and the direct comparison test. These tests involve comparing the given integral to a known integral that can be evaluated, and using the properties of limits to determine the convergence or divergence of the given integral.
Yes, an improper integral with variable can have an infinite limit of integration. In this case, the integral will need to be evaluated as a limit as the variable approaches infinity or negative infinity. This is often seen in integrals involving trigonometric functions or exponential functions.
There are some special cases and exceptions for evaluating improper integrals with variables, such as integrals with infinite limits of integration or integrals with integrands that have vertical asymptotes within the interval of integration. In these cases, alternative techniques such as partial fractions or integration by parts may be used to evaluate the integral.