# 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 x^{2} is 2x, the indefinite integral of 2x is given by:

$$ \int 2x \ dx = x^2+c $$

Here, x^{2} + c represents the family of all antiderivatives of 2x.

To illustrate, the function x^{2}+10 is an antiderivative of 2x because when differentiated, D[x^{2} + 10] = 2x. Similarly, x^{2}-5 is another antiderivative of 2x as D[x^{2}-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:

$$ \int 2x \ dx = x^2+c $$

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