Why can the dx in an integral be treated as a differential?

[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 I am making some kinda sense here.

 

[tmc]

you mean something like this?

<br /> \[<br /> \begin{array}{l}<br /> y = 3x^2 \\ <br /> \frac{{dy}}{{dx}} = 6x \\ <br /> dy = 6xdx \\ <br /> \int {dy} = \int {6xdx} \\ <br /> y = 3x^2 \\ <br /> \end{array}<br /> \]<br />

 

[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.

 

[Zurtex]

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.

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.