Definition:Improper Integral

Definition
An improper integral is a definite integral over an interval which is not closed, that is, open or half open, and whose limits of integration are the end points of that interval.

When the end point is not actually in the interval, the conventional definition of the definite integral is not valid. Therefore we use the technique of limits to specify the integral.

Note: In the below, in all cases the necessary limits must exist in order for the definition to hold.

Unbounded Open Intervals
The same techniques can be modified for unbounded open intervals in the forms $\left({a \,.\,.\, +\infty}\right)$ and $\left({-\infty \,.\,.\, b}\right)$:

Notation
It is common practice to remove the $\to$ sign from the limits of integration, for example: $\displaystyle \int_{-\infty}^{+\infty} f \left({t}\right) \, \mathrm dt$.

However, this is not recommended, as confusion can result, in particular when investigating Lebesgue integration.