Improper integral

An improper integral is a term from the mathematical subfield of Analysis. With the help of this integral term, it is possible to integrate functions that have singularities or individual whose domain is unrestricted, and therefore can not be integrated as such. The improper integral can be understood as an extension of the Riemann integral, the Lebesgue integral, or other integration concepts. Often it is however considered in connection with the Riemann integral, since especially the ( actual ) Lebesgue integral can integrate already many features that are only improperly Riemann integrable.

• 2.1 Two fractional rational functions
• 2.2 Gaussian error integral

Definition

There are two reasons why improper integrals are considered. On the one hand one would like to integrate functions via unrestricted areas, for example up. This is the Riemann integral is not readily possible. Improper integrals that solve this problem are called improper integrals of the first kind Moreover, it is also of interest to integrate functions that have a singularity on the boundary of its domain. Improper integrals, which make this possible are called improper integrals of the second kind, it is possible that improper integrals of the first kind at a boundary improperly and on the other border of the second kind are inauthentic. However, it is irrelevant to the definition of the improper integral, of what sort it is integral.

Integration area with a critical limit

Be and a function. So the improper integral in the case of convergence is defined by

Similarly, the improper integral for and defined.

Integration area with two critical limits

Be and a function. So the improper integral in the case of convergence is defined by

Being considered and the two right integrals improper integrals with a critical limit is. Advertised means

And the convergence of the value of the integral does not depend on the choice of.

Examples

Two fractional rational functions

If a primitive is known, as in the actual case, the integral is evaluated at the adjacent location, and then the limit value for calculated. An example is the integral of

In which the integrand has a singularity at and therefore does not exist as a ( real ) Riemann integral. Summing up the integral as improper Riemann integral of the second kind, so true

The integral

Has an unrestricted domain and is therefore an improper integral of the first kind applies

Gaussian error integral

The Gaussian error integral

Is an improper Riemann integral of the first kind in the sense of Lebesgue integration theory, the integral exists in the real sense.

Relationship between actual and improper Riemann and Lebesgue integrals

• Each Riemann - integrable function is Lebesgue integrable.
• Thus, any improper Riemann - integrable function is also improper Lebesgue integrable.
• There are functions that are improperly Riemann integrable but not Lebesgue integrable, consider about the integral

• On the other hand, there are functions that are Lebesgue integrable, but are not improperly Riemann integrable, consider this as the Dirichlet function on a bounded interval.