An improper integral is an integral that has infinity in one of its “extremities”.

\[\int_a^\infty f(x)dx = \lim_{b \to \infty} \int_a^b f(x)dx\]

This means that the upper limit of this integral is the limit that \(b\) approaches as \(b\) goes to infinity. If the limit exists, we say tha the improper integral converges, otherwise, we say that it diverges.

Two more type of improper integrals follow from the first definition.

\[\int_{-\infty}^b f(x)dx = \lim_{a \to -\infty} \int_a^b f(x)dx\]

\[\int_{-\infty}^\infty f(x)dx = \int_{-\infty}^c f(x)dx + \int_{c}^{\infty} f(x)dx\]