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1Necessary to make a whole complete; essential or fundamental.

*‘games are an integral part of the school's curriculum’**‘systematic training should be integral to library management’**‘He was integral to the whole process and I'll probably vote for him in the final round of voting.’**‘Musically, it often has a really boring part but that part is integral to the chord structure.’**‘He says one of those ‘bite your thumb’ lines so integral to the whole adolescent love story.’**‘The branches are not just appendages; they are integral to the whole.’**‘A telecoms worker, for example, is both productive in providing a vital service and is also integral to the handling of information.’**‘He has the speed, patience and quick burst necessary to make the cutback runs that are integral to the offense.’**‘It is also rooted in a people's culture, and integral to their structure of thought.’**‘‘I recognise that church and community halls are integral to community life and provide a social hub in a great many rural and urban areas,’ said Mr Howells.’**‘He said: ‘This is integral to us bringing the jobs we have promised so of course we want to see it go ahead.’’**‘Reputations aside, looking the part has certainly been integral to this almost episodic drama, at least as it's unfolded in the media.’**‘In 1989 John Williamson asked Pixie to back him in his live shows, and for the next five years, Pixie became an integral part of John's live performances and recordings.’**‘Producer John Simon was an integral part of the Band's classic recordings.’**‘An integral part of any call center is its recording system: it is used for training agents, ensuring quality across a campaign, legal compliance and more.’**‘Limousines, luxurious hotels, fancy restaurants and trips to the recording studio are all an integral part of a popstar's life.’**‘Records are an integral requirement for organic certification.’**‘Since 1921 artistic, photographic and written records have formed an integral part of the Air Force's heritage.’**‘They seem to be thrown in as an afterthought, rather than as an integral portion of the story.’**‘Its core principle is that the teacher is integral not separate from his or her teaching, making the experience holistic.’**‘Thus, they serve more as an accompaniment rather than an integral part of the book.’**‘Logistic companies now form an integral part of the supply chain of any efficient business.’*

View synonyms**essential**, fundamental, basic, intrinsic, inherent, constitutive, innate, structural- 1.1attributive Included as part of a whole rather than supplied separately.
*‘the unit comes complete with integral pump and heater’**‘The integral power supply may be backed up by an external 24-volt DC supply, ensuring continuity of programming and easy mobile use.’**‘A future mission should treat a Mars lander as an integral part of the whole spacecraft rather than one of its instruments, the report said.’**‘The best option is either a pressurised system or a power shower, which is a mixer shower with an integral pump.’**‘I have a humongous bay window of gigantic proportions and my new curtains, once made up, will have cost me the slim sum of only £20 which includes integral linings.’**‘Set over three floors, including an integral garage, this newly built three-bedroom house is in Kemp Town, close to all of Brighton's action.’**‘Exclusive detailing includes integral roof rails, a rear screen that opens independently of the tailgate.’**‘First, since they're luxury seats, they have a lot of integral features including power lumbar control, seatheaters, and side impact airbags.’**‘Included is an integral belt, skid-plate chest covering, face/rifle scope veil and elbow/knee pads.’**‘The barrel length is 19.8 inches including the well-designed integral muzzle brake.’**‘Scope rings to fit the integral receiver grooves are included with each T3.’**‘Outside, the property includes a double integral garage, a paddock, extensive lawns and a vegetable garden.’**‘It also includes an integral altitude warning system to avoid deviations.’**‘Of the various firearms which include an integral lock, I know of none as neat and unobtrusive.’**‘This then forms the cover, which includes both an integral hinge and a tear seam.’**‘The unit has an integral vacuum system that incorporates a turbomolecular pump as well as a backing scroll pump.’**‘Heat pump water heaters can be purchased as integral units with built in water storage tanks or as add-ons that can be retrofitted to an existing water heater tank.’**‘As a result, many major manufacturers are equipping their firearms with integral locking devices or including trigger locks with every firearm.’**‘The same has been known to happen with the small keys that operate gun locks, whether they are separate or integral.’**‘This results in a heavier, stronger frame and also allows the use of an integral scope mounting system on the frame rather than the barrel.’**‘Formerly a hotel, it has six en-suite bedrooms, four other bedrooms, an integral garage and, unusually, a recording studio.’*

View synonyms**built-in**, inbuilt, integrated, incorporated, fitted, component, constituent - 1.2attributive Having or containing all parts that are necessary to be complete.
*‘the first integral recording of the ten Mahler symphonies’**‘Now, of course, the composer appears on even major labels with some regularity, and there have been several integral recordings of the symphonies.’*

View synonyms**unified**, integrated, comprehensive, organic, composite, combined, aggregate, undivided, overall, gross, entire, complete, whole, total, full, intact

2Mathematics

Of or denoted by an integer.*‘Other topics he worked on include algebraic geometry, number theory and integral equations.’**‘He developed the relation between the algebra of matrices and integral equations as well as studying boundary value problems.’**‘He was particularly interested in the courses in complex variable, integral equations and differential equations.’**‘Under his guidance she worked on integral equations studying infinite dimensional linear spaces.’**‘Another analysis topic he studied was non-linear integral equations.’*

- 2.1Involving only integers, especially as coefficients of a function.
*‘To do this we make adjustments in the integral functions.’**‘Barnes next turned his attention to the theory of integral functions, where, in a series of papers, he investigated their asymptotic structure.’**‘He received his doctorate for a thesis entitled 'Contributions to the theory of integral functions of finite order' in 1929.’**‘While he was doing this wide range of work he was also proving some of his deepest results in the study of integral functions.’**‘Some of them are about the theory of equations, others about integral functions.’*

1Mathematics

A function of which a given function is the derivative, i.e. which yields that function when differentiated, and which may express the area under the curve of a graph of the function.See also definite integral, indefinite integral*‘In Appendix B, the formula for calculating the integral of a logistic curve is given.’**‘It describes the integral of the area and the angular extents over which a radiation transfer problem is defined.’**‘Both the peak values and the integrals under the characteristic fluorescence curves were measured.’**‘Hence an awareness of the inverse of differentiation began to evolve naturally and the idea that integral and derivative were inverses to each other were familiar to Barrow.’**‘I remember feeling this way about derivatives (or was it integrals?).’*

- 1.1A function satisfying a given differential equation.
*‘He wrote on algebraic integrals of certain differential equations.’**‘His first mathematical research was on analysis, in particular concentrating on integrals and solutions of differential equations.’**‘His work on algebra was an extension of Abel's work on algebraic functions and their integrals.’**‘He showed how to find integrals of a general system of partial differential equations by using sequential complete systems instead of passing to Jacobian systems.’**‘An estimate of an integral of a function can be obtained with a Monte Carlo integration.’*

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