Is the Integral Syntax Correct for dx = v dt and x = \int v dt?

[konik]

Is the following syntax correct?

$$dx = v\ dt$$
$$x = \int v\ dt$$

or should it be:

$$dx = v \ dt$$
$$dx = \int v \ dt$$

 

[Doc Al]

Is the following syntax correct?

$$dx = v\ dt$$
$$x = \int v\ dt$$

This is OK. Realize that this is just saying that:
$$\int \ dx = x$$

or should it be:

$$dx = v \ dt$$
$$dx = \int v \ dt$$

That's no good--you must integrate both sides.

 

[HallsofIvy]

The first. Obviously, the two right sides of the second are not the same and cannot both be equal to dx.

What you are doing is starting with dx= v dt and integrating both sides:
$\int x= \int v dt$. Since $\int dx= x$ the result is $x= \int v dt$.

(The integral is not "well defined" so that should be $x= \int v dt+ C$.)
(Once again, Doc Al comes in 2 seconds ahead of me!)

 

[Gib Z]

The integral is not "well defined" so that should be $x= \int v dt+ C$

We don't really need to include the additional C, since indefinite integrals are only unique up to a additive constant anyway.

 

[HallsofIvy]

Yes, of course. The anti-derivative of $x^2$ is $\int x^2 dx$ which is, itself, equal to $(1/3)x^3+ C$. It is only in the last that we need the "C".

 

[CompuChip]

Actually, I prefer to think of
$$\int v \, dt$$
as notation where the $\int dt$ is a single symbol.
The equation
$$dx = v dt$$
wouldn't make sense then but can be written
$$dx/dt = v$$
or considered as a limit.

(Of course, I also use them as mnemonics and manipulate them as ordinary fractions, but sometimes it's good to keep things clear and separate legal operations from convenient notation).