antiderivatiivne

Antiderivatiivne is a term used in mathematics, specifically in calculus, to refer to the concept of an antiderivative. An antiderivative of a function f(x) is a function F(x) whose derivative is f(x). In other words, if F'(x) = f(x), then F(x) is an antiderivative of f(x).

The process of finding an antiderivative is called antidifferentiation or integration. For any given function, there are infinitely many antiderivatives. These antiderivatives differ from each other by a constant. If F(x) is an antiderivative of f(x), then F(x) + C is also an antiderivative of f(x) for any constant C. This is because the derivative of a constant is always zero, so the derivative of F(x) + C is F'(x) + 0, which is still f(x). The collection of all antiderivatives of a function is called the indefinite integral and is denoted by the integral symbol ∫ f(x) dx = F(x) + C.

Antiderivatives are fundamental to solving differential equations and calculating definite integrals. The Fundamental Theorem of Calculus establishes a crucial link between differentiation and integration, stating that the definite integral of a function can be evaluated by finding an antiderivative and computing the difference of its values at the limits of integration. For example, an antiderivative of f(x) = 2x is F(x) = x², because the derivative of x² is 2x. Therefore, the indefinite integral of 2x is ∫ 2x dx = x² + C.