Antiderivatives and indefinite integration

Indefinite integration, also known as antidifferentiation, is the reversing of the process of differentiation. Given a function f, one finds a function F such that F' = f.

Finding an antiderivative is an important process in calculus. It is used as a method to obtain the area under a curve and to obtain many physical and electrical equations that scientists and engineers use everyday. For example, the equation for the current through a capacitor is

I = Cdv/dt, where I is current in Amperes, C is capacitance in Farads, V is voltage in Volts and t is time in seconds. To obtain an unknown (like V), one would have to use integration to obtain a voltage at a certain time interval.

While a true integral exists between a given boundary, taking the indefinite integral is simply reversing differentiation in much the same way division reverses multiplication. Instead of having a set of boundary values, one only finds an equation that would produce the integral due to differentiation without having to use the values to get a definite answer.

We use the notation F(x) = to indicate that f is the indefinite integral of f. By using this notation F(x) = if and only if F`(x) = f(x).

Properties of indefinite integrals

Uniqueness theorem :

If F and G are antiderivatives of some interval I, then there is a constant C such that F(x) = G(x) + C for all x in I. As a consequence of this theorem, we will usually add c to an indefinite integral.

Inverse property of indefinite integrals

In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i.e., F` = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite function is called differentiation, which is the process of finding a derivative. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

The discrete equivalent of the notion of antiderivative is antidifference.