Etymology
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Words related to integrate

in- (1)
Origin and meaning of in-
word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."

In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
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*tag- 
Proto-Indo-European root meaning "to touch, handle," with figurative extensions ("border on; taste, partake of; strike, hit; affect, impress; trick, cheat; mention, speak of").

It forms all or part of: attain; contact; contaminate; entire; intact; integer; integrate; integrity; noli me tangere; tact; tactics; tactile; tangent; tangible; task; taste; tax; taxis.

It is the hypothetical source of/evidence for its existence is provided by: Latin tangere "to touch," taxare "to touch, assess," tactus "touch," integer "intact, whole, complete, perfect; honest;" Greek tassein "to arrange," tetagon "having seized;" Old English þaccian "stroke, strike gently."
integration (n.)
1610s, "act of bringing together the parts of a whole," from French intégration and directly from Late Latin integrationem (nominative integratio) "renewal, restoration," noun of action from past participle stem of Latin integrare "make whole," also "renew, begin again" (see integrate). Anti-discrimination sense (opposed to segregation) is recorded from 1934.
integrated (adj.)
1580s, "combined into a whole," past-participle adjective from integrate (v.). Sense of "desegregated, not or no longer divided by race, etc." is from 1947.
disintegrate (v.)

1796, transitive, "separate into component parts, destroy the cohesion of," originally in geology, from dis- "do the opposite of" + integrate (v.). Intransitive sense, "to break apart, separate into its component parts," is by 1851. Related: Disintegrated; disintegrating.

reintegrate (v.)

c. 1600, "renew with regard to any state or quality," from re- "back," here "to a former condition," + integrate (v.). The sense of "make whole again, bring back to an integral condition" is from 1620s. The classically correct form is redintegrate (early 15c.). Earlier in a now-obsolete sense of "reinstate oneself" (1580s). Related: Reintegrated; reintegrating; reintegration.