antideriválokkal

An antiderivative, also known as an indefinite integral, is a function whose derivative is a given function. If F(x) is an antiderivative of f(x), then the derivative of F(x) with respect to x is equal to f(x), which can be written as F'(x) = f(x). The process of finding an antiderivative is called antidifferentiation or integration.

The antiderivative of a function is not unique. If F(x) is an antiderivative of f(x), then any function of the form F(x) + C, where C is an arbitrary constant, is also an antiderivative of f(x). This is because the derivative of a constant is zero, so the derivative of F(x) + C is F'(x) + 0 = f(x). This arbitrary constant is often denoted by "+ C" when writing indefinite integrals.

Antiderivatives are fundamental to calculus and have numerous applications. They are used to solve differential equations, which model many phenomena in science and engineering. They are also used to calculate definite integrals, which represent areas under curves, volumes of solids, and accumulated change. 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 endpoints of the interval. Basic rules exist for finding antiderivatives of common functions, such as power functions, trigonometric functions, and exponential functions, often derived by reversing the differentiation rules.