> [!note] Definition > > $ > \begin{align*} > \int_{a}^{\infty}f(x)\ dx &= \lim_{t \to \infty}\int_{a}^{t} f(x)\ dx \\ > \int_{-\infty}^{a}f(x)\ dx &= \lim_{t \to -\infty}\int_{t}^{a} f(x)\ dx \\ > \int_{-\infty}^{\infty}f(x)\ dx &= \int_{-\infty}^{a}f(x)\ dx + \int_{a}^{\infty} f(x)\ dx \\ > \end{align*} > $ > > An improper integral is the [[Limit|limit]] of a [[Definite Integral|definite integral]] as the endpoint of the bounds approach some number, either because the bounds approach infinity, or because there is a [[Continuity|discontinuity]] on the integrand at the bounds.