Antiderivative
The antiderivative of a function, denoted as f(x), is another function F(x) such that the derivative F'(x) equals f(x).
$$ F'(x)=f(x) $$
This antiderivative is also referred to as the indefinite integral, represented as:
$$ \int f (x) \ dx = F(x) + c $$
The indefinite integral of f(x) does not represent a singular function F(x), but rather a family of functions described by F(x) + c , where "c" is an arbitrary constant.
Consider the following example: since the derivative of x2 is 2x, the indefinite integral of 2x is given by:
$$ \int 2x \ dx = x^2+c $$
Here, x2 + c represents the family of all antiderivatives of 2x.
To illustrate, the function x2+10 is an antiderivative of 2x because when differentiated, D[x2 + 10] = 2x. Similarly, x2-5 is another antiderivative of 2x as D[x2-5] = 2x, and so forth.
Distinction Between Antiderivative and Definite Integral
While the symbols for the indefinite (antiderivative) and definite integrals appear similar, they convey distinct concepts.
The definite integral includes two numbers, termed the limits of integration, and computes the area beneath the function's graph. For instance:
$$ \int_1^4 2x \ dx = (4)^2 - (1)^2 = 16 - 1 = 15 $$
Conversely, the indefinite integral yields a function:
$$ \int 2x \ dx = x^2+c $$