What makes an integral improper?
Improper integrals are a kind of definite integral, in the sense that we're looking for area under the function over a particular interval. This is in opposition to an indefinite integral, where we're looking for a function that represents the area everywhere under the function.
Sometimes people think that integrals are only defined as improper when the limits of integration are infinite (meaning that they are positive or negative infinity). While it's true that these are considered improper integrals, there are actually multiple ways that an integral can qualify as improper.
Integrals are improper when either the lower limit of integration is infinite, the upper limit of integration is infinite, or both the upper and lower limits of integration are infinite.
An integral is also considered improper if the integrand is discontinuous on the interval of integration, which means that the function we're integrating has a discontinuity in the interval. The discontinuity can also occur at either of the endpoints of the interval, or at both endpoints, and the function would still be considered to have a discontinuity in the interval, and would therefore still be an improper integral.
In this video we'll discuss exactly what it takes to make an integral improper.