Improper Riemann Integrals

Riemann integral is defined only for bounded functions defined on a set of compact intervals.

Type 1

A function that is not integrable at one endpoint of the interval.

Suppose is integrable on .

Can be similarly defined on the other endpoint. The above integral converges iff the limit exists and finite. Otherwise diverges.

Examples

The above integral converges (to ) iff . When , it diverges to .

Type 2

A function defined on unbounded interval (including ).

Suppose is integrable on .

Can be similarly defined on the other endpoint. The above integral converges iff the limit exists and finite. Otherwise diverges.

Examples

The above integral converges (to ) iff . When , it diverges to .

Type 3

A function that is undefined at finite number of points. The integral can be split into multiple integrals of type 1. Similarly integrals from to can be defined.

Note

The integral can be split into multiple integrals only when all those integrals exist.

Convergence of improper integrals is similar to the convergence of series.

Absolute convergence test