integranden

Integranden, in calculus, is the term for the function that is being integrated in an integral. In the notation ∫_a^b f(x) dx, f(x) is the integrand; in an indefinite integral, ∫ f(x) dx, the integrand is likewise the function being integrated. For multiple integrals, such as ∫∫_D f(x,y) dx dy, the integrand is f(x,y). Integranden may depend on parameters, yielding parameterized integrals f(x; θ).

Most common properties: If the integrand is continuous on a closed interval [a,b], then it is Riemann integrable on that interval. More generally, a bounded function with a set of discontinuities of measure zero is Riemann integrable. The Lebesgue integral extends integrals to a wider class of integrands, including functions with unbounded or many discontinuities, by measuring the size of level sets.

Operations and transformations: The integral is linear: ∫ (α f + β g) = α ∫ f + β ∫ g. If a substitution or change of variables is performed, the integrand changes accordingly; in definite integrals, the integrand reflects the function to be averaged or accumulated. In cases of singularities or infinite limits, an integral can be defined as a limit of integrals over finite ranges (improper integrals).

Examples: The integrand in ∫_0^1 x^2 dx is x^2. In ∫_a^b e^{−x^2} dx, the integrand is e^{−x^2}. The integrand may be more complex or involve several variables, such as f(x,y) in a double integral.

Applications: Integranden arise in calculations of areas, probabilities, expected values, physics, and engineering, where the integral computes a quantity accumulated from the integrand over a domain.