B Undefined Integration

1. Jul 25, 2017

Aows

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

Last edited by a moderator: Jul 25, 2017
2. Jul 25, 2017

Staff: Mentor

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.

3. Jul 25, 2017

Aows

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

4. Jul 25, 2017

Staff: Mentor

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.

5. Jul 25, 2017

Aows

i need a reason why it is undefined.

6. Jul 25, 2017

Staff: Mentor

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 $"\cdot" \, dx$

Last edited: Jul 25, 2017
7. Jul 25, 2017

Staff: Mentor

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