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 $\hointr a \to$.
Then the improper integral of $f$ over $\hointr a \to$ is defined as:
- $\ds \int_a^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to +\infty} \int_a^\gamma \map f t \rd t$
Also denoted as
It is common to abuse notation and write:
- $\ds \int_a^\infty \map f t \rd t$
which is understood to mean exactly the same thing as $\ds \int_a^{\mathop \to + \infty} \map f t \rd t$.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definition of a Definite Integral: $15.3$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.27$