integral
/ˈɪntɪɡrəl/ (bre, ipa) · /ˈɪntɪɡrəl/ (ame, ipa) · /ˈin-ti-grəl (usually so in mathematics) in-ˈte-grəl also -ˈtē- also nonstandard ˈin-trə-gəl/ (ame, mw) · /ˈɪn.tɪ.ɡrəl/ (bre, ipa) · /ˈɪn.t̬ə.ɡrəl/ (ame, ipa)
integral — adjective
- integralpositive
- more integralcomparative
- most integralsuperlative
1. belonging to something in such a central way that the whole thing cannot functio
belonging to something in such a central way that the whole thing cannot function, exist, or be complete without it
The software is now an integral part of the school's daily teaching system.
collocation: integral part of
Good communication skills were integral to Asher's success as a team leader.
pattern: integral + to + noun
Exercise and a healthy diet are integral parts of a strong recovery plan.
The new public library became integral to the life of the local community.
Trust is integral to any strong friendship between two people.
- essential
the closest synonym; both suggest something cannot be removed without damaging the whole
- crucial
emphasises importance for success or outcome; slightly stronger than 'integral'
- fundamental
stresses being at the base or foundation of something
- vital
implies something is needed for life or continued existence; stronger than 'integral'
- peripheral
describes something that is on the edge and not central
- inessential
means something can be removed without affecting the whole
文法句型
integral + to + noun phrase
integral + part of + noun phrase
用法筆記
Often followed by 'to' to show what something is essential for. Can also appear before 'part of' — 'an integral part of something' is a common fixed expression.
常見錯誤
2. designed or built into something as one unit, rather than being added on or supp
designed or built into something as one unit, rather than being added on or supplied as a separate item
The new car has an integral GPS system built right into the dashboard.
collocation: integral + noun (built-in feature)
In this design, the handle is integral to the frame, so you cannot remove it.
pattern: integral to + noun
The speakers are integral to the television, not separate add-on devices.
Takeshi's kitchen has an integral waste-disposal unit under the sink.
Modern smartphones have an integral camera that cannot be detached.
- built-in
more common in everyday speech; 'integral' is slightly more formal
- inbuilt
British English variant; same meaning as 'built-in'
- incorporated
suggests something has been deliberately included as part of a design
- separate
describes parts that are not physically joined and can be removed
- detachable
means designed to be taken off or disconnected
- external
means located outside the main body or unit
文法句型
integral + noun
integral + to + noun phrase
用法筆記
Commonly describes parts or features of machines, devices, or buildings that are made as one piece rather than added later. Distinguish from sense 1 (ESSENTIAL PART), where the focus is on importance; here the focus is on physical or structural unity.
常見錯誤
3. containing every part that is needed, with nothing missing; complete and entire
containing every part that is needed, with nothing missing; complete and entire
The museum proudly presented its integral pottery collection after finally acquiring the missing Ming vase.
attributive: integral + noun for complete collection
The historian managed to reconstruct the integral document from pieces found in three different archives.
After finding the last rare issue, Élise finally owned an integral stamp collection spanning every period.
Adina's integral set of antique coins now includes every year from the series without exception.
The orchestra performed the integral symphony, including all four movements.
- incomplete
describes something that is missing one or more parts
- partial
means only part of the whole is present
文法句型
integral + noun
become + integral
be + integral
用法筆記
This sense is formal and less frequent than senses 1 and 2. It is often used in academic writing, art criticism, or historical descriptions to emphasise that a collection or text has all its parts. Distinguish from sense 2 (BUILT-IN), which describes parts joined together physically; sense 3 describes a whole thing whose all parts are present.
integral — noun
- integralsingular
- integralsplural
1. a number or formula that represents the total value of a changing quantity over
a number or formula that represents the total value of a changing quantity over a certain range, especially used in calculus to calculate areas under curves and volumes of solids
In calculus class, the students learned how to calculate the integral of a simple quadratic function.
collocation: calculate the integral
The definite integral of the function between x = 1 and x = 4 gives the area under the curve.
collocation: definite integral
Adina used a computer program to find the integral of the equation.
To find the volume of the solid, the engineer first had to solve a double integral.
The integral of velocity over time equals the total distance travelled.
- antiderivative
technical synonym for indefinite integral; less common in basic calculus courses
- definite integral
a specific type of integral that produces a numerical value over a defined range
- derivative
the inverse operation in calculus; a derivative measures the rate of change, whereas an integral measures accumulation
文法句型
the + integral + of + noun phrase
definite/indefinite + integral
calculate/find/solve + the + integral
用法筆記
A technical term in calculus. The two main types are the 'definite integral' (which gives a number) and the 'indefinite integral' (which gives a family of functions). Not to be confused with 'integer', which is a whole number.