# Antiderivative

An **antiderivative** of a function f(x) is another function F(x) such that F'(x)=f(x). The antiderivative is also called the **indefinite integral**. $$ \int f (x) \ dx = F(x) + c $$ The indefinite integral of f(x) is not a single function F(x), it is a family of all antiderivatives F (x) + c of the function f(x). The term c is any constant.

For example, becaus the derivative of x^{2} is 2x, the indefinite integral of 2x is x^{2}+c

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

The solution x^{2}+c is a family of all antiderivatives of 2x.

**Verify**. The function x^{2}+10 is an antiderivative of 2x because D_{x}[x^{2}+10]=2x. The function x^{2}-5 is another antiderivative of 2x because D_{x}[x^{2}-5]=2x and so on

### Difference between anti-derivative and definite integral

The symbol of the indefinite integral (anti-derivative) is similar to that of the definite integral but expresses a different concept.

The definite integral contains two numbers called __limits of integration__ and calculates the area under the graph of the function.

$$ \int_1^4 2x \ dx = (4)^2 - (1)^2 = 16 - 1 = 15 $$

Vice versa, the indefinite integral is a function.

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