n. 1. (for a given function, f(x)) the limit, evaluated by the integral calculus, of the sum of the rectangular elements f(x)δx, where δx is a subinterval of a partition of an interval of values of the independent variable, and the limit is taken as the number of subintervals tends to infinity and the length of each tends to zero. This is sometimes expressed in terms of an infinite sum of infinitesimal quantities. The area, lightly shaded in the figure below, between the curve y = f(x) and the x-axis, and between x = a and x = b is the limit of the sum of such rectangular elements of area, of which one is represented by the darker shaded rectangle in the figure above, whose bases are a partition of [a, b]; this is the definite integral of f(x) from a to b, and is written An indefinite integral or antiderivative, written is any other function of x whose derivative is f(x), and is unique up to a constant; for example, the indefinite integral of axⁿ is where c is a constant; however, this constant of integration is often omitted, and this practice has been followed in the list of common indefinite integrals in Appendix 2. Integrals are formally defined in terms of upper and lower Darboux sums; and the definite and indefinite integrals are related by the fundamental theorem of calculus. These concepts can be extended by the iterated integral to multiple integration. See also Riemann integral, Lebesgue integral.
The integral from a to b is the limit of the sum of the elements f(x)δx.
2a. the symbolic representation of a definite or indefinite integral.
b. the symbol ∫. 3. a solution of a differential equation. 4. adj. of or pertaining to integers. An integral polynomial is a polynomial with rational integer coefficients.