An Antiderivative or indefinite integral of a function is a differentiable function
whose derivative
is equal to
, i.e.
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 function is an antiderivative of
. As the derivative of a constant
is zero, x2 will have an infinite number of antiderivatives; such as
etc.
Thus, all the antiderivatives of can be obtained by changing the value of
in
;
where
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 .
The simplest case, the integral over of a real-valued function
, is written as
.
The integral sign represents integration. The
indicates that we are integrating over
;
is
called the variable of integration.