Indefinite integration
In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F ′ = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation 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.
The function F(x) = x3/3 is an antiderivative of f(x) = x2, as the derivative of x3/3 is x2. As the derivative of a constant is zero, x2 will have an infinite number of antiderivatives, such as x3/3, x3/3 + 1, x3/3 - 2, 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 vertical location depending upon the value of C.
In physics, the integration of acceleration yields velocity plus a constant. The constant is the initial velocity term that would be lost upon taking the derivative of velocity because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion (position, velocity, acceleration, and so on).
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