Antiderivaadad

Antiderivaadad, also known as primitive functions, are a fundamental concept in calculus. An antiderivative of a function f(x) is a function F(x) whose derivative is f(x). In simpler terms, if you differentiate F(x), you get back f(x). This relationship is the inverse of differentiation.

The process of finding an antiderivative is called antidifferentiation or integration. For any given function f(x), there are infinitely many antiderivatives. This is because the derivative of a constant is always zero. Therefore, if F(x) is an antiderivative of f(x), then F(x) + C is also an antiderivative, where C is an arbitrary constant. This constant C is known as the constant of integration.

The indefinite integral of f(x), denoted by the integral symbol ∫ followed by f(x) and dx, represents the collection of all antiderivatives of f(x). So, ∫ f(x) dx = F(x) + C. The dx symbol indicates that the integration is with respect to the variable x.

Antiderivatives are crucial for solving differential equations, calculating areas under curves (definite integrals), and in various applications across physics, engineering, and economics. Understanding the relationship between differentiation and antidifferentiation is key to mastering calculus.