MHB What is the integral of dx over dx?

[Evgeny.Makarov]

What is $\displaystyle\int\frac{dx}{dx}$? There is a reasonable answer to this question.

Also, does anybody know what this integral is called?

 

[mvCristi]

I respect your question (and you), but I think you cannot speak about dx/dx, unless you're some "story teller, spelling in mathematical simbols" callculus pseudoexpert wannabe. (I'm very sure you're not such a person. ;) )

 

[Evgeny.Makarov]

To be sure, the question is a little

, but there is a perfectly reasonable answer.

 

[mvCristi]

As "falsehood implies everything, it implies every reasonable answer." But I'm not polemic! What 1/dx means? Integral(dx/dx) is lacking specific entities, such as a function (like the identity function). It is usually understood from the context. But just make an entire description. You may use Weirstarass Calculus or the HyperReal Numbers of Newton and Cauchy. Just state it clearly and I will answer it.

 

[Plato2]

What is $\displaystyle\int\frac{dx}{dx}$? There is a reasonable answer to this question. Also, does anybody know what this integral is called?

@Evgeny.Makarov
I will certainly yield to you on any question of logic.
But here is my research area.
Those of us in the H.S.Wall/Gilliam tradition have no idea what $\int f $ means,
We know what $\int_a^b f $ means.

Sorry to say, I find your question meaningless.

 

[Evgeny.Makarov]

Well, OK. For $\displaystyle\int\frac{dx}{dx}$ to make sense, the expression after the integral sign must have a single $dx$ in the nominator. Therefore, $dx$ in the denominator is not a differential, but rather a product of a constant $d$ and a variable $x$. Therefore, $\displaystyle\int\frac{dx}{dx}=\frac{\ln|x|}{d}+C$.

Edit: And why is it that you know what $\int_a^bf$ means but don't know what $\int f$ means?

 

[Plato2]

For $\displaystyle\int\frac{dx}{dx}$ to make sense, the expression after the integral sign must have a single $dx$ in the nominator. Therefore, $dx$ in the denominator is not a differential, but rather a product of a constant $d$ and a variable $x$. Therefore, $\displaystyle\int\frac{dx}{dx}=\frac{\ln|x|}{d}+C$.

Sorry, but as an member of what is generally call the "Texas school" I find that a meaningless expressionism.
I understand that logicians take liabilities in defining notations.
But I am not ready to give you free rain with this this one.

x And why is it that you know what $\int_a^bf$ means but don't know what $\int f$ means?

Some of us have had to give into publishers in order of get textbooks out.
That simply means anti-derivative.
Anti-derivative is not integral,

 

[Evgeny.Makarov]

All this is supposed to be a sort of a joke. That's why I referred to "tongue-in-cheek" in post #3.

I understand that logicians take liabilities in defining notations.

This question was told to me by fellow students probably during my first or second year, even before I chose to specialize in logic.

I think also that lateral thinking used here can be useful not so much in mathematics, but in programming, where programs sometimes parse not according to common sense. I don't have a good example right now; if I come up with one, I'll post it here.

That simply means anti-derivative.
Anti-derivative is not integral,

Then I think $\int f$ should denote the antiderivative and $\int^b_a f$ should not make sense.

 

[mvCristi]

It's tricky the solution. And I did not expected to have such a simple answer.

For an example of how programs parse do you refer to something like this (in C):
(0 && (b=getch()), where if some compiling time optimizations are set, the program will not read b?

/* let's say we know a*a+b*b>0 */
/* let's say we are satisfied with a very rough approximation*/
#define infinity 4294967295
if (( a = getValue() ) && (( b = aVeryResourceIntensiveComputation() ))
{ c = b/a; }
else
{ c = infinity;}
makeSomeVeryRoughApproximation(c);

 

[Plato2]

Then I think $\int f$ should denote the antiderivative and $\int^b_a f$ should not make sense.

If you can find the first edition of Calculus by Gillman&McDowell you will see that in fact the notation
$\int^b_a f$ is used for an integral.

Gillman is known as a stickler for correctness of notation. After all he was brought to Texas to fill the void created by the forced retirement of R.L. Moore.

If you can find an actual copy of that text, you will see my pick for the best ever calculus textbook. The size of the book is totally reasonable; the typography is beautiful; there is no need for technology.

Most importantly, Gillman develops the integral by way of a betweeness property.

As a side bar: I think that Gillman is the only MAA president who was also a Julliard graduate.