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.

Examples

Half Open Intervals

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

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

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

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

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:

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

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

Unbounded Half Open Intervals

• Let $f$ be a real function which is continuous on the unbounded half open interval $\left[{a .. \infty}\right)$.

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

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

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

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:

$\displaystyle \int_{\to -\infty}^{\to +\infty} f \left({t}\right) dt := \lim_{\gamma \to -\infty} \int_\gamma^0 f \left({t}\right) dt + \lim_{\gamma \to +\infty} \int_0^\gamma f \left({t}\right) dt$

Notation

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

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

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 13.27$