integral function
C2Academic/Technical
Definition
Meaning
A mathematical function that expresses the integral (antiderivative) of another function; specifically, a function F(x) whose derivative is equal to the original function f(x).
In calculus, a function whose derivative is a given function. More generally, in mathematics and engineering, any function defined by an integration process, such as a cumulative distribution function or an error function. The term can also refer to a function that is itself defined as an integral, like the gamma function.
Linguistics
Semantic Notes
The phrase is a noun-noun compound primarily used in mathematics. 'Integral' functions as an adjective modifying 'function,' specifying the type of function. Avoid confusion with 'integral' as an adjective meaning 'essential' (e.g., 'an integral part'). In mathematics, it's strictly related to integration.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling of related terms may differ (e.g., 'antiderivative' is common in both, but British texts may use 'primitive function' more frequently).
Connotations
Purely technical with identical connotations in both varieties.
Frequency
Used with comparable frequency in academic mathematics and engineering contexts in both the UK and US.
Vocabulary
Collocations
Grammar
Valency Patterns
The integral function of [mathematical expression]Find the integral function for/of [function name]Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Not used.
Academic
Core term in calculus, real analysis, differential equations, and engineering mathematics.
Everyday
Not used in everyday conversation.
Technical
Essential in physics, engineering, econometrics, and any field involving continuous modelling and integration.
Examples
By CEFR Level
- The student struggled to find the integral function for the complex equation.
- In physics, velocity is the integral function of acceleration with respect to time.
- The existence of an elementary integral function for this rational expression is not guaranteed by Liouville's theorem.
- One must consider the constant of integration when defining the general form of an indefinite integral function.
Learning
Memory Aids
Mnemonic
Think: INTEGral Function = INTEGration's result Function. It's the function you get from integrating (accumulating) another function.
Conceptual Metaphor
RECONSTRUCTION FROM RATE OF CHANGE (The integral function reconstructs the total quantity from its rate of change, like finding distance travelled from a speedometer reading).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid confusing 'integral function' with 'integral' as a noun meaning a definite integral value (число). 'Integral function' is a функция.
- Do not translate 'integral' in this context as 'целостный' or 'неотъемлемый'. It is strictly 'интегральный' or related to 'интегрирование'.
Common Mistakes
- Using 'integral function' to mean 'essential function' in non-mathematical contexts.
- Confusing 'integral function' (antiderivative) with 'definite integral' (a number/area).
- Incorrect stress: placing primary stress on the second syllable of 'integral' (/ɪnˈtɛɡ.rəl/) is less common for the mathematical adjective in this compound.
Practice
Quiz
Which of the following best describes an 'integral function'?
FAQ
Frequently Asked Questions
Essentially, yes. The indefinite integral of f(x) is the family of all integral functions (antiderivatives) of f(x), differing only by a constant.
Yes, if the original function being integrated has discontinuities, its integral function may be continuous but not necessarily differentiable everywhere.
An integral function (or antiderivative) is itself a function. A definite integral is a numerical value representing the area under a curve between two limits.
No. Not all functions have an antiderivative expressible in terms of elementary functions (e.g., e^(-x^2)). Furthermore, functions must meet certain conditions (like being continuous on an interval) to guarantee the existence of an integral function.