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.

Half Open Intervals

 * Let $f$ be a real function which is continuous on the half open interval $\left[{a \,.\,.\, b}\right)$.


 * $\displaystyle \int_a^{\to b} f \left({t}\right) \, \mathrm dt := \lim_{\gamma \to b} \int_a^\gamma f \left({t}\right) \, \mathrm dt$


 * Let $f$ be a real function which is continuous on the half open interval $\left({a \,.\,.\, b}\right]$.


 * $\displaystyle \int_{\to a}^b f \left({t}\right) \, \mathrm dt := \lim_{\gamma \to a} \int_\gamma^b f \left({t}\right) \, \mathrm dt$

Open Intervals

 * Let $f$ be a real function which is continuous on the open interval $\left({a \,.\,.\, b}\right)$.

In this situation, there are two limits to consider.

A useful technique here is to split the open interval up into two half open intervals.

Let $c \in \left({a \,.\,.\, b}\right)$, thus $\left({a \,.\,.\, b}\right) = \left({a \,.\,.\, c}\right] \cup \left[{c \,.\,.\, b}\right)$.

Then:


 * $\displaystyle \int_{\to a}^{\to b} f \left({t}\right) \, \mathrm dt := \lim_{\gamma \to a} \int_\gamma^c f \left({t}\right) \, \mathrm dt + \lim_{\gamma \to b} \int_c^\gamma f \left({t}\right) \, \mathrm dt$

The validity of this approach is justified by Sum of Integrals on Adjacent Intervals‎.

Unbounded Closed Intervals

 * Let $f$ be a real function which is continuous on the unbounded closed interval $\left[{a \,.\,.\, +\infty}\right)$.


 * $\displaystyle \int_a^{\to + \infty} f \left({t}\right) \, \mathrm dt := \lim_{\gamma \to + \infty} \int_a^\gamma f \left({t}\right) \, \mathrm dt$


 * Let $f$ be a real function which is continuous on the unbounded closed interva $\left({-\infty \,.\,.\, b}\right]$.


 * $\displaystyle \int_{\to -\infty}^b f \left({t}\right) \, \mathrm dt := \lim_{\gamma \to -\infty} \int_\gamma^b f \left({t}\right) \, \mathrm dt$

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

A specific and important instance of this occurs when the interval in question is the set of all real numbers.


 * Let $f$ be a real function which is continuous everywhere.

Then:


 * $\displaystyle \int_{\to -\infty}^{\to +\infty} f \left({t}\right) \, \mathrm dt := \lim_{\gamma \to -\infty} \int_\gamma^\lambda f \left({t}\right) \, \mathrm dt + \lim_{\gamma \to +\infty} \int_\lambda^\gamma f \left({t}\right) \, \mathrm dt$

where $\lambda$ is any constant real number, but it usually taken as $0$ for simplicity.

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.