Antiderivatives

An antiderivative of a function f is another function F whose derivative equals f. In symbols, F'(x) = f(x). Finding an antiderivative is the reverse operation of differentiation. The set of all antiderivatives of f is F(x) + C, where C is an arbitrary constant.

The notation ∫ f(x) dx is used to denote an antiderivative, called an indefinite integral. Because differentiating F(x) + C yields f(x) for any constant C, the constant of integration accounts for all possible antiderivatives.

The Fundamental Theorem of Calculus links differentiation and integration. If f is continuous on an interval and F is any antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a). This theorem shows that differentiation and integration are inverse processes on such intervals.

Common examples illustrate simple antiderivatives. The antiderivative of x^n (with n ≠ −1) is x^{n+1}/(n+1) + C. The antiderivative of 1/x is ln|x| + C. The antiderivative of e^x is e^x + C.

Applications of antiderivatives include computing areas under curves, accumulating quantities, and solving problems in physics and engineering. Not every function has an antiderivative that can be expressed in elementary functions; some require special functions or numerical methods.