Improper integrals involve integrating functions with infinite or discontinuous limits from "summary" of Skills in Mathematics - Integral Calculus for JEE Main and Advanced by Amit M Agarwal
Improper integrals are a special type of integrals where the limits of integration involve infinity or points where the function being integrated is discontinuous. These integrals do not fit into the standard definition of integrals, which deal with functions that are continuous over a finite interval. In the case of improper integrals, the limits of integration may extend to infinity, which means the function is being integrated over an unbounded interval. When dealing with improper integrals, it is important to understand that traditional methods of integration may not always apply. The presence of infinite or discontinuous limits requires a different approach to evaluating the integral. In some cases, the function being integrated may not be defined at certain points within the interval of integration, which further complicates the calculation process. One common example of an improper integral is the integral of a function that has a vertical asymptote within the interval of integration. In such cases, the function is discontinuous at the point of the asymptote, making it impossible to directly evaluate the integral using standard methods. To handle this situation, special techniques such as limit calculations or integration by parts may be required. Another scenario where improper integrals arise is when the function being integrated approaches infinity at one or both of the limits of integration. In these cases, the integral may not converge to a finite value, which poses a challenge in determining the area under the curve. Techniques such as comparison tests or convergence tests are often employed to determine whether the integral converges or diverges.- Improper integrals present unique challenges due to their infinite or discontinuous nature. Understanding how to handle these types of integrals is crucial for tackling complex integration problems that involve functions with unconventional limits of integration. By employing specialized techniques and methods, it is possible to evaluate improper integrals and obtain meaningful results in the realm of integral calculus.
