Integrally

Integrally is an adverb derived from the adjective integral, which itself comes from Latin integralis meaning whole or complete. In general English, integrally describes something that is joined, unified, or functioning as a single unit. It is commonly used to stress wholeness or inseparability of parts, as in "the components are integrally linked" or "the system operates integrally with its surrounding infrastructure." The term is frequent in engineering, manufacturing, architecture, and design, where components may be formed integrally (as a single piece) rather than assembled from separate elements.

In mathematical discourse, the primary terms are integral and integration. While "integral" refers to the calculus concept or to something essential, "integrally" is less common in formal mathematics. It may appear in broader or interdisciplinary discussions to convey a sense of wholeness or a relationship to the concept of integration, but it is not a standard technical term in calculus.

Etymology: integrally derives from Latin integralis, from integer "whole." The word entered English through Old French and Middle English usage.

Usage notes: prefer integrally to express unity, coherence, or completeness of parts. When referring to calculus or the mathematical integral, use "integral" or phrases like "in integral form" or "in terms of the integral."

See also: integral, integrate, integrality.