Primitive, antiderivative and integral

[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

?

 

[mathman]

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).

 

[Jhenrique]

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.

 

[gopher_p]

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.

 

[mathman]

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.