Antidérivées

An antiderivative, also known as an indefinite integral, is a function whose derivative is a given function. If F is an antiderivative of f, then the derivative of F with respect to x is equal to f(x). The concept is fundamental to calculus, as it represents the inverse operation of differentiation.

For any given function f, there are infinitely many antiderivatives. These antiderivatives differ from each other by a constant. If F is an antiderivative of f, then F + C is also an antiderivative of f, where C is an arbitrary constant. This is because the derivative of a constant is always zero. Therefore, the general form of the antiderivative of f is denoted by F(x) + C.

The notation for the indefinite integral of a function f(x) with respect to x is the integral symbol ∫ followed by f(x) dx. So, if F'(x) = f(x), then ∫f(x) dx = F(x) + C. Finding antiderivatives is often referred to as integration. Certain rules and techniques exist to find antiderivatives for various types of functions, such as power rule, constant multiple rule, sum and difference rule, and substitution.

The existence of antiderivatives is guaranteed for continuous functions over an interval. This is stated by the fundamental theorem of calculus, which establishes a deep connection between differentiation and integration. The first part of the theorem states that if f is continuous on [a, b] and F(x) = ∫a^x f(t) dt, then F'(x) = f(x). This formally proves that continuous functions have antiderivatives.