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