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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from Calculus: Help and ReviewChapter 8 / Lesson 12