Integral Notation: Are ∫(x^2)dx & ∫dx (x^2) the Same?

[DiracPool]

I'm confused over two different types of integral notation

1) ∫ (expression) dx

and

2) ∫dx (expression)

Are these the same thing?

Example: Do ∫(x^2)dx and ∫dx (x^2) mean the same thing? Or is there a difference?

 

[pwsnafu]

Mathematicians use (1), physicists use (2). For all intents and purposes they mean the same.

 

[DiracPool]

Great. Thanks.

 

[Fredrik]

I'd like to change "physicists use (2)" to "some physicists use (2)" or "physicists occasionally use (2)".

I think the idea behind (2) is that it mimics the d/dx notation for derivatives. When you write $\frac{d}{dx}ax^2$, the d/dx is like an operator that acts on $ax^2$. Actually, it acts on the map $x\mapsto ax^2$, not the real number $ax^2$, and it's the x in the denominator of d/dx that let's us know that the map is $x\mapsto x^2$ rather than say $a\mapsto x^2$.

Similarly, $\int dx\, ax^2$ is $\int dx$ acting on $ax^2$, or to be more precise, on the map $x\mapsto ax^2$, and now it's the x in dx that tells us that the map is $x\mapsto ax^2$ rather than e.g. $a\mapsto ax^2$.

 

[DivergentSpectrum]

a little history : the ∫ sign is an elongated S, which stands for sum because its the sum of all values of expression. multiply that by dt, and you get the integral
so both are valid, but personally i always use ∫f(x)dx because it tells you where the expression ends.
edit: its so typical of physicists to do this kind of stuff :P