Integrovat

Integrovat is a Czech verb meaning to perform the operation of integration, typically in a mathematical context. It is derived from the noun integral (integrál) and the concept of integration (integrace). In mathematics, integrovat a function means to find its integral, either analytically or numerically.

There are two main notions: indefinite and definite integrals. An indefinite integral, written as ∫ f(x) dx, represents a family of antiderivatives F(x) + C, where F'(x) = f(x). A definite integral, written as ∫_a^b f(x) dx, assigns a numerical value often interpreted as the accumulated quantity such as area under a curve between a and b. The Fundamental Theorem of Calculus links these notions: if F is an antiderivative of f, then d/dx ∫_a^x f(t) dt = f(x), and ∫_a^b f(x) dx = F(b) − F(a).

Techniques for analytic integration include substitution (u-substitution), integration by parts, partial fractions, trigonometric substitutions, and the use of special functions. When an antiderivative cannot be expressed in elementary terms, numerical integration methods are used, such as the trapezoidal rule, Simpson’s rule, Romberg integration, or Gauss quadrature.

Applications of integrovat and integration span many fields: calculating areas, determining probabilities and expected values, solving problems in physics (work and energy), economics, and engineering. In Czech mathematical language, the activity is described as integrovat funkci, with the integral referred to as integrál; related terms include integrace and integrovat se (to integrate oneself into a system or process) in broader contexts.