Integrands

An integrand is the function that is being integrated in an integral expression. In the notation ∫ f(x) dx, the function f(x) is the integrand and the variable of integration is x. Integrands can be functions of one variable or several variables, and they may appear in definite integrals, indefinite integrals, or more general forms such as line, surface, or contour integrals.

Definite integrals evaluate the accumulation of quantities over a domain, with the integrand determining what is being accumulated. Indefinite integrals represent families of antiderivatives: if F is an antiderivative of f, then ∫ f(x) dx = F(x) + C, and the same integrand f appears in both forms.

Conditions for integrability depend on the integration framework. In Riemann integration on a closed interval, the integrand must be bounded and have a finite number of discontinuities (or more generally, the set of discontinuities has measure zero). In Lebesgue integration, the integrand must be measurable and either nonnegative or absolutely integrable; the integral is defined as an extended real number. Some integrands are improper, requiring limits at endpoints or singularities.

Examples and notes: common integrands include polynomials, exponentials, trigonometric functions, and rational functions. In multivariable calculus, integrands may depend on several variables and appear under double or triple integrals, with the differential element reflecting the domain of integration (dx, dy, dA, dV). In more advanced contexts, integrands can be vector- or matrix-valued and appear inside line, surface, or contour integrals. The choice of integrand determines the interpretation of the integral as area, accumulated quantity, probability, or other aggregate measures.