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 x2 is 2x, the indefinite integral of 2x is x2+c

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

The solution x2+c is a family of all antiderivatives of 2x.

Verify. The function x2+10 is an antiderivative of 2x because Dx[x2+10]=2x. The function x2-5 is another antiderivative of 2x because Dx[x2-5]=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.

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

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