Differentials under differentials in integrals

[Signifier]

Hello.

Are the following integrals equivalent:

Integral from 0 to 5 of dx / x

and

Integral from 0 to 5 of dx / (x + dx)

What about

Integral from 0 to 5 of dx / (x + 2dx + dx^2)

?

If they are all equivalent, why? (I have an intuitive answer, but it has 0 mathematical foundation). Or are they not all the same? Can someone explain this strongly?

Thank you!

 

[quasar987]

Take note that you should not take my answer seriously, but I am tempted to propose an answer, and see how people react to it.

The way I would interpret

$$\int_0^5 \frac{dx}{x+dx+(dx)^2}$$

is by treating the dx on top as the dx that's part of the notation for "the integral of a function", and the rest of them I would treat as constant (or more precisely, as parameters), so the answer would be

$$\ln (5+dx+(dx)^2)-ln(dx+(dx)^2)$$With this interpretation, you can see that your three integrals are not equal.

 

[Hurkyl]

I would say that it's nonsense unless you defined what it meant.


Boldly doing manipulations without any solid justification (which is the only kind we can do when we don't have a definition for what we're manipulating)...

$$\frac{dx}{x + dx + (dx)^2} = \frac{1}{x + dx + (dx)^2} dx \approx \frac{1}{x} \left(1 - \frac{dx + (dx)^2}{x}\right) \, dx \approx \frac{dx}{x}$$

 

[Signifier]

I am interested primarily because of a curious step in this derivation of the Tsiolkovsky "rocket equation": http://ed-thelen.org/rocket-eq.html

Namely, where he says, "we are looking at the area under the curve of 1/(x+dx) where in the process we make dx arbitrarily small so that we are looking at the area under the curve of 1/x in the limit. In some presentations, this simplification is done before the integration is performed."