In calculus, an "anti-derivative", antiderivative, primitive integral 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.
Rules and formulae
Antidifferentiation is the process of finding the set of all antiderivatives of a given function. The symbol denotes the operation of antidifferentiation
The expression F(x) + C is the general antiderivative of ƒ. The constant C is used because the derivative of a constant is always 0. The notation C should be understood to mean a quantity that is constant on every connected interval within the domain separately, but not necessarily constant on the whole domain. For example
Because antidifferentiation is the inverse operation of the differentiation, antidifferentiation theorems and rules are obtained from those on differentiation. Thus, the following theorems can be proven from the corresponding differentiation theorems:
- General antidifferentiation rule:
- The general antiderivative of a constant times a function is the constant multiplied by the general antiderivative of the function:
- If ƒ and g are defined on the same interval, then the general antiderivative of the sum of ƒ and g equals the sum of the general antiderivatives of ƒ and g:
The function F(x) = x3/3 is an antiderivative of f(x) = x2. As the derivative of a constant is zero, x2 will have an infinite number of antiderivatives; such as (x3/3) + 0, (x3 / 3) + 7, (x3 / 3) − 42, etc. Thus, all the antiderivatives of x2 can be obtained by changing the value of C in F(x) = (x3 / 3) + C; where C is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other; each graph's location depending upon the value of C.
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