Antiderivate

An antiderivative of a function f is another function F whose derivative is f; that is F'(x) = f(x) for all x in a given interval. The antiderivative is also called an indefinite integral, denoted ∫ f(x) dx, and it represents a family of functions differing only by a constant.

The antiderivative is not unique: if F is an antiderivative of f, then F + C is also an antiderivative for any constant C.

The fundamental theorem of calculus relates differentiation and integration: if f is continuous on [a, b] and F is an antiderivative of f on that interval, then ∫_a^b f(x) dx = F(b) − F(a).

Common examples: the antiderivative of x^2 is x^3/3 + C, the antiderivative of e^x is e^x + C, and the antiderivative of sin x is −cos x + C.

Techniques for finding antiderivatives include the power rule, substitution, integration by parts, partial fractions, and trigonometric substitutions. Some functions do not have elementary antiderivatives and are handled with special functions or numerical methods.

Antiderivatives are used to compute areas, accumulate quantities, and solve differential equations, among other applications in mathematics, physics, and engineering.