Definition:Improper Integral/Unbounded Closed Interval/Unbounded Above

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Definition

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

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

$\displaystyle \int_a^{\mathop \to + \infty} \map f t \rd t := \lim_{\gamma \mathop \to +\infty} \int_a^\gamma \map f t \rd t$


Also denoted as

It is common to abuse notation and write:

$\displaystyle \int_a^\infty \map f t \rd t$

which is understood to mean exactly the same thing as $\displaystyle \int_a^{\mathop \to + \infty} \map f t \rd t$.


Sources