> [!NOTE] Definition > Let $a\in \mathbb{R}.$ If $f:[a,\infty)\to \mathbb{R}$ is a [[Real Function|real function]] that is [[Riemann integration|integrable]] on each $[a,b]$ for all $b\geq a,$ then the improper integral of $f$ over $[a,\infty)$ is given by the [[Limit of Real Function at a Point|limit]] $ \int_{a}^{\infty} \, dx = \lim_{ b \to \infty } \int_{a}^{b} f(x) \, dx$ > and we say that $f$ is improperly Riemann integrable on $[a,\infty)$ the limit exists. >