What is the correct notation for plugging in 0 to an anti-derivative?

[Cinitiator]

Homework Statement

I need to plug 0 into an anti-derivative of f, and I'm wondering whether it can be written as:

$y = \int f(0)dx$

or do I HAVE to write it as:
$F(x) =\int f(x)dx$
$y = F(0)$

Homework Equations

See 1.

The Attempt at a Solution

See 1.

 

[lanedance]

neither really, as the variable you integrate over is treated as a dummy variable, so really something like below would be better
$$F(x)=\int_{x_0}^{x}f(s)ds$$

the x0 point determines your constant of integration

 

[Cinitiator]

neither really, as the variable you integrate over is treated as a dummy variable, so really something like below would be better
$$F(x)=\int_{x_0}^{x}f(s)ds$$

the x0 point determines your constant of integration

Why can't it just be:
$F(x) + C =\int f(x)dx$

$y = F(0) + C$

With "C" being the constant of integration.

 

[Fredrik]

Homework Statement

I need to plug 0 into an anti-derivative of f, and I'm wondering whether it can be written as:

$y = \int f(0)dx$

or do I HAVE to write it as:
$F(x) =\int f(x)dx$
$y = F(0)$

When I see the expression $\int f(0) dx$, I'm thinking that f(0) is a constant that can be taken outside of the integral. The notation $F(x)=\int f(x)dx$ is no better, because the left-hand side depends on x (unless F is a constant function), while the right-hand side doesn't. So the equality doesn't define a function, even if you specify that it holds for all x.

Why not just say something like

Let F be any function such that F'=f.​

or

Let F be the unique function such that F'=f and F(0)=C.​

?
Then you can denote the value of F at 0 by F(0).

Lanedance's suggestion is also good.

 

[lanedance]

yeah just to add the integral depends on only on the endpoints of the integral, and the form of the function.

when you have f(x) and sat the antiderivative is F(x) + c, the easiest way to think of it is the constant is related to where the integral started (x0) whilst x represents the end of the integration interval. In performing the integration, f takes all possible values on the interval (x0,x)

 

[Cinitiator]

Does one have to add the constant of integration (C) when using the 'from 0 to x' definite integral method to express indefinite integrals at a given x?

Ex:
$$F(x)=\int_{0}^{x} f(t)dt$$

I still have to add "C" to F(x), am I right?

 

[Fredrik]

Does one have to add the constant of integration (C) when using the 'from 0 to x' definite integral method to express indefinite integrals at a given x?

Ex:
$$F(x)=\int_{0}^{x} f(t)dt$$

I still have to add "C" to F(x), am I right?

If the problem you're trying to solve asked you to find all antiderivatives of f, then yes. The F defined by the formula above satisfies F'=f and F(0)=0, so it's just one of the infinitely many antiderivatives of f. However, note that $\int_a^x f(t)dt$ is equal to $\int_0^x f(t)dt$ plus a constant that depends on a. So each choice of a gives you a different antiderivative of f.