Antiderivat

An antiderivative of a function f is another function F such that F' = f on an interval. It is the inverse operation of differentiation and is also called a primitive or an indefinite integral. It is denoted ∫ f(x) dx, with the addition of a constant of integration, since derivatives of constants vanish.

The Fundamental Theorem of Calculus links antiderivatives and definite integrals: if F is an antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) − F(a). This theorem provides a bridge between the accumulation of quantities and the rate at which they change.

Examples: F(x) = x^2/2 is an antiderivative of f(x) = x, since F'(x) = x. ∫ e^x dx = e^x + C, ∫ sin x dx = −cos x + C, and ∫ 1/x dx = ln|x| + C for x ≠ 0. Some functions do not have antiderivatives that can be expressed in elementary form, such as ∫ e^(−x^2) dx, which is related to the erf function and is often evaluated numerically.

Techniques to find antiderivatives include substitution (u-sub), integration by parts, and partial fractions. Not all integrals have elementary antiderivatives; in such cases numerical methods or special functions are used.

Applications of antiderivatives include computing areas, solving problems involving accumulation, and addressing differential equations with given initial conditions. The indefinite integral expresses a family of antiderivatives through the constant C.