What is the name for F(x) and ##\mathcal{F}(x)## in this context?

[Jhenrique]

When we took the integral of a function f, the result is: $$\int f(x)dx = F(x) + C$$ But, F(x) + C can be rewritten like: $$F(x) + C = \mathcal{F}(x)$$ So, my first ask is: which is the name given for $F(x)$ and for $\mathcal{F}(x)$ ? I can't call both of primitive of f, because it's confuse. I already heard the term "family of antiderivative", I think that this term is the name of $\mathcal{F}(x)$, thus, maybe, primitive is better for $F(x)$. What do you think?

---

Second question: by FTC: $$\int_{x_0}^{x}f(x)dx = F(x) - F(x_0)$$ implies that: $$F(x) = \int_{x_0}^{x}f(x)dx + F(x_0)$$ This result, F(x), represents the $F(x)$ or the $\mathcal{F}(x)$ of the my 1st question?

---

OBS: if you affirm that F(x) of the 2nd question is equal to $\mathcal{F}(x)$ this implies that $\int_{x_0}^{x}f(x)dx = F(x)$ and $F(x_0)=C$ .

 

[Mark44]

When we took the integral of a function f, the result is: $$\int f(x)dx = F(x) + C$$ But, F(x) + C can be rewritten like: $$F(x) + C = \mathcal{F}(x)$$

What's the point? Is there some purpose in writing F(x) + C as $\mathcal{F}(x)$?

So, my first ask is: which is the name given for $F(x)$ and for $\mathcal{F}(x)$ ?

F is an antiderivative of f. I've also seen it called a primitive.

I can't call both of primitive of f, because it's confuse. I already heard the term "family of antiderivative", I think that this term is the name of $\mathcal{F}(x)$, thus, maybe, primitive is better for $F(x)$. What do you think?

---

Second question: by FTC: $$\int_{x_0}^{x}f(x)dx = F(x) - F(x_0)$$ implies that: $$F(x) = \int_{x_0}^{x}f(x)dx + F(x_0)$$ This result, F(x), represents the $F(x)$ or the $\mathcal{F}(x)$ of the my 1st question?

F is any antiderivative of f (or any primitive of f).

---

OBS: if you affirm that F(x) of the 2nd question is equal to $\mathcal{F}(x)$ this implies that $\int_{x_0}^{x}f(x)dx = F(x)$ and $F(x_0)=C$ .

 

[Jhenrique]

There is a importnat difference between $F(x)$ and $\mathcal{F}(x)$, the 1st is a particular solution for $\int f(x) dx$ and the 2nd is general solution. By be different solutions, maybe they have different names. How I already heard a lot times the term "antiderivative" and "family of antiderivative", I asked if those terms are the names for $F(x)$ and for $\mathcal{F}(x)$ ...

EDIT: YEAH! An teacher confirmed my hypothesis above.

 

[Mark44]

There is a importnat difference between $F(x)$ and $\mathcal{F}(x)$, the 1st is a particular solution for $\int f(x) dx$ and the 2nd is general solution.

I disagree. The expression F(x) + C doesn't represent a particular antiderivative unless C is somehow specified to be a particular value. In that sense (i.e., C being an unspecified arbitrary value) F(x) + C represents the entire family of antiderivatives. Again, I still don't see the point of writing both F(x) + C and $\mathcal{F}(x)$. IMO it's much ado about a minor point.

By be different solutions, maybe they have different names. How I already heard a lot times the term "antiderivative" and "family of antiderivative", I asked if those terms are the names for $F(x)$ and for $\mathcal{F}(x)$ ...

EDIT: YEAH! An teacher confirmed my hypothesis above.

I still disagree.
For example, consider $\int x^2 dx$.
One antiderivative is (1/3)x³ + 7.
All antiderivatives have the form (1/3)x³ + C.