# What is the difference between anti-derivatives and integrals?

## Question:

What is the difference between anti-derivatives and integrals?

## Antiderivative of a Function:

The process to get the anti-derivative of a function *f(x*) consists in looking for another function *F(x)* which is the primitive function of *f(x),* this means that when we compute the derivative of the function *F(x*) it gives us the function *f(x)*, in other words, *F'(x)=f(x). *The indefinite integral of that function *f(x)* would give as a result *F(x)+C* in which the letter *C* indicates the value of an arbitrary constant.

## Answer and Explanation:

The anti-derivative (also called primitive) is a function that when we derive it we get the original function. Example:

*F(x)=6x* is an anti-derivative of* f(x)=6*

As we see a show has a lot of anti-derivatives. In the previous example we can see that the functions *F(x)=6x+1, F(x)=6x-3*, *F(x)=6x+7, F(x)=6x-12*, ... are also anti-derivatives of *f(x)=6.*

As we can see, when calculating the derivative *F(x)* we obtain *f(x)*.

On the other hand, we have that the indefinite Integral represents the set of all the infinite quantity of primitives that the function *f(x) *can have. That is, the indefinite integral of a function is equal to its antiderivative plus a constant, following the previous example:

*F(x)=6x+C* is the indefinite integral of *f(x)=6* since the constant *C* includes all arbitrary constants.

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