# Improper integrals

A definite integral is an integral between two finite points *a* and *b*:

We normally expect *f* to be continuous and bounded over the interval *[a, b]*.

It is sometimes possible to break these conditions. Integrals that break these conditions are called *improper integrals*, but in some cases it is quite possible to calculate the area under the curve. Here are three examples:

The first is:

This integral extends to infinity, but as *x* tends to infinity, the function tends to zero, and we will show that the area under the curve converges to a finite value.

The second is:

The final integral is:

This integral goes from minus infinity to infinity. As *x* tends to infinity in either direction, the function tends to zero, and again we will show that the area under the curve converges to a finite value.