# Primitive, antiderivative and integral

• Jhenrique
In summary: The mathematical problem is the same regardless.In summary, "primitive", "antiderivative" and "integral" are often used interchangeably, with the only real distinction being between definite and indefinite integrals. The Fundamental Theorem of Calculus relates indefinite integrals to antiderivatives, but the terms are fundamentally different objects.
Jhenrique
Prays the CFT that all function f(x) can be expressed how the integral of its derivative more an initial constant:$$f(x)=\int_{x_0}^{x}f'(u)du+f(x_0)$$ So, is correct affirm that integral, primitive and antiderivative are concepts differents? ie:

f(x) = primitive

##\int_{x_0}^{x}f'(u)du## = antiderivative

∫ = integral

?

Last edited:
Jhenrique said:
Prays the CFT that all function f(x) can be expressed how the integral of its derivative more an initial constant:$$f(x)=\int_{x_0}^{x}f(u)du+f(x_0)$$ So, is correct affirm that integral, primitive and antiderivative are concepts differents? ie:

f(x) = primitive

##\int_{x_0}^{x}f(u)du## = antiderivative

∫ = integral

?

I am not sure what you are asking. However:
$$f(x)=\int_{x_0}^{x}f(u)du+f(x_0)$$ should be:
$$F(x)=\int_{x_0}^{x}f(u)du+F(x_0)$$

where f(x) = F'(x). F(x) is the antiderivative (indefinite integral) of f(x).

Actually, I wrong the notation, but I already fix!

So, in other words, I'd like to know if exist difference between "primitive", "antiderivative" and "integral", if thoses concepts are the same thing or if they are differents. This is my question.

As far as i can tell, "primitive" and "antiderivative" are, apart from some minor quibbles regarding whether you're talking about a function or a family of functions, synonymous (they are different words that refer to the same concept). For most of the people that I know, "indefinite integral" is also synonymous with "primitive" and "antiderivative", though the indefinite integral always (in my experience) denotes a family of functions.

There are some folks who call a function of the form ##F(x)=\int\limits_a^xf(t)\ dt## an "indefinite integral", though I believe this usage is very uncommon in modern undergraduate classrooms/texts. In this case "indefinite integral" is not synonymous with "primitive" and "antiderivative". It is fundamentally a different object, though the Fundamental Theorem of Calculus does give us a way to relate it to antiderivatives; in particular it tells us that ##F## is an antiderivative of ##f## whenever ##f## is a derivative.

I'm not sure what you are asking. However, making a distinction between these terms is quibbling. The only real distinction is between definite integral and indefinite integral.

Your example is that of a definite integral being called indefinite because the upper limit is a variable. It is OK as long as it is clear to everyone what is being done.

## 1. What is the difference between a primitive and an antiderivative?

A primitive is a function that, when differentiated, gives the original function. An antiderivative is a function that, when differentiated, gives the original function plus a constant.

## 2. How do you find the primitive of a given function?

To find the primitive of a given function, you need to reverse the process of differentiation. This can be done by using the power rule, product rule, quotient rule, and chain rule in reverse. Additionally, there are certain common functions that have well-known primitives, such as the natural logarithm and exponential functions.

## 3. What is the relationship between a primitive and an integral?

The integral of a function is the area under the curve of that function. The primitive of a function is the inverse process of differentiation. It can be thought of as the area function, where the value of the primitive at a given point is the area under the curve from the starting point of the integration to that point.

## 4. Can all functions have a primitive?

No, not all functions have a primitive. For a function to have a primitive, it must be continuous on its entire domain. Additionally, some functions, such as those with discontinuities or vertical asymptotes, may have primitives but may require special techniques to find them.

## 5. How is the concept of integration related to real-world applications?

The concept of integration is used in many real-world applications, such as calculating the area under a curve in physics or finding the total volume of a three-dimensional object in engineering. It is also used in economics to calculate total revenue or profit, and in statistics to find the probability of certain events. In essence, integration helps us understand and analyze continuous quantities in various fields.

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