An improper integral is a definite integral in which one or both of the limits is infinite, or where the function being integrated becomes infinite at some point within the limits.

For example, \[\begin{align*} \int_{-1}^1 \frac{1}{x^2} \:dx, \\ \int_0^\infty e^{-x} \:dx \end{align*}\]

are both improper.

In some cases, an improper integral can be evaluated fairly straightforwardly. In other cases, it is not possible.