Differentials in Integrals

1. Dec 6, 2005

G01

I was just wondering about the dx and the end of an integral and evaluating integrals by substitution. When you evaluate integrals by substitution you can treat the dx as the differential of x. This seems to convenient lol. Some one must ahve known that the dx in an integral was the differential of x all along when they decided to end the integral symbol with the term dx! Could someone please give me insight into this. Jeeze I hope Im making some kinda sense here.:tongue2:

2. Dec 6, 2005

tmc

you mean something like this?

$$$\begin{array}{l} y = 3x^2 \\ \frac{{dy}}{{dx}} = 6x \\ dy = 6xdx \\ \int {dy} = \int {6xdx} \\ y = 3x^2 \\ \end{array}$$$

3. Dec 6, 2005

G01

yeah but i don't need to know how they work but why you can treat the dx in an integral as a differential. I know it sounds weird bear with me.

4. Dec 6, 2005

Zurtex

Well generally, when you learn basic calculus such as that is stated above, without the integral sign dx on its own it's just lazy notation and not rigorously defined. It's just a bit of time and hides a bunch of results in it such as the Fundamental Theorems of Calculus:

http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html

However, dx can actually be rigorously defined on its own and is frequently used in such subject matter as Vector Calculus.