An improper integral is an integral where one or both of the limits of integration are infinite, or the integrand is undefined at one or more points in the interval.
Improper integrals pose a problem because they do not satisfy the basic conditions for Riemann integrability, which requires the function to be bounded and continuous on a closed interval. This can lead to unexpected and sometimes undefined results.
To determine if an improper integral converges or diverges, you can use the limit comparison test, comparison test, or the integral test. These tests evaluate the behavior of the integral as the limits of integration approach infinity or negative infinity.
No, improper integrals cannot be solved using traditional integration methods like the power rule or substitution. Instead, they require specialized techniques such as partial fractions, trigonometric substitutions, or breaking the integral into smaller, convergent parts.
Improper integrals are used in many real-world applications, such as calculating areas under unbounded curves, finding the center of mass of an object, and determining the probability of continuous events in statistics. They also have uses in physics, engineering, and economics.