antiderivative

An antiderivative (or indefinite integral, \[\int f(x)\,dx\]) of a continuous function \[f\] is a differentiable function \[F\] whose derivative is equal to the original function \[f\], or \[F^{\prime}=f\].

Antiderivatives are related to definite integrals through the second fundamental theorem of calculus, as the definite integral of a function over a closed interval is the difference between the values of an antiderivative evaluated at the endpoints of the interval, i.e. \[\int_{a}^{b}f(x)\,dx=(F(b)+c)-(F(a)+c)=F(b)-F(a)\].

Generally, \[\int f(x)\,dx=F(x)+c\], as a consequence of the first fundamental theorem of calculus (\[F^{\prime}(x)=f(x)\]), and the fact that the derivative of a constant, \[c\], is zero (therefore \[\frac{d}{dx}F(x)=\frac{d}{dx}\left[ F(x)+c \right]=f(x)\]). Therefore, it follows that \[f(x)\] has an infinite number of antiderivatives, as \[c\in \mathbb{R}\].