# Integral Syntax Question

1. Dec 29, 2007

### konik

Is the following syntax correct?

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

or should it be:

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

2. Dec 29, 2007

### Staff: Mentor

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

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

3. Dec 29, 2007

### 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!)

4. Dec 29, 2007

### Gib Z

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

5. Dec 30, 2007

### 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".

6. Dec 30, 2007

### 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).

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