Definition:Improper Integral/Unbounded Open Interval/Unbounded Above

Definition

Let $f$ be a real function which is continuous on the unbounded open interval $\openint a {+\infty}$.

Then the improper integral of $f$ over $\openint a {+\infty}$ is defined as:

$\ds \int_{\mathop \to a}^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to +\infty} \int_c^\gamma \map f t \rd t$

for some $c \in \openint a {+\infty}$.