Why is integration with dx in the denominator undefined?

[Aows]

Hello,
In your opinion, why the integration with dx in the denominator is undefined ?? (as in the attached picture)

 

[fresh_42]

Hello,
In your opinion, why the integration with dx in the denominator is undefined ??

Could you explain what you mean? What do you think does dx have to do in the integral?
This question is a bit like asking why ${}^25$ is undefined.

 

[Aows]

Ok, here is the question in clearer way (as in the attached pic)

 

[fresh_42]

This is undefined, because nobody defined it. And nobody defined it, because there is no need to define it. At least no mathematical need. One could attach any meaning to it, but this would cause confusions with the integral $\int \frac{1}{x}\,dx$ which is defined.

 

[Aows]

I can't understand your answer.
i need a reason why it is undefined.

 

[fresh_42]

I can't understand your answer.
i need a reason why it is undefined.

What reason do you need and why? A definition is an action. In order to act, people need a motivation. But there is none. It is that simple. Why isn't $\int x\,\clubsuit \,x$ defined? I try to figure out, whether you had something in mind in order to want to define it, or you just picked some symbols and modeled a question with it.

The point is, that $\int f(x)dx$ isn't really a "multiplication". $dx$ is rather a symbol to indicate the variable of integration. Because it is no multiplication, why should there be a division? Of course it is related to a multiplication as we use integrals to calculate areas, which are width times length. So width divide by length is the slope of something, which leads to differentiations, where $\frac{d}{dx}$ is denoted as a quotient. But this isn't a true division either, only an encoded text which describes what really has to be done. So a notation $"\colon" \, dx$ leads to differentiation which is the correspondence to the integration $"\cdot" \, dx$

 

[Mark44]

Thread closed. As @fresh_42 already said, writing an integral with a differential in a denominator is not defined.