# Differentials under differentials in integrals

Hello.

Are the following integrals equivalent:

Integral from 0 to 5 of dx / x

and

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

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
Homework Helper
Gold Member
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.

Last edited:
Hurkyl
Staff Emeritus
Gold Member
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)...

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

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."