Sometimes I see that
$\int f(x) dx = \int f(x) *dx $
This property is required by u substitution like so: $ \int f(g(x)) dx = \int \frac {f(u)}{u'} du$
or some fancy integrals like so: $ \int x^{dx} -1 = \int \frac {x^{dx} - 1}{dx} dx$
this property is also used in differential equations and so on.
However I fail to see the rigor behind this. Because dx in this context is defined to be the variable of the anti derivative.
I tried to prove this through riemann sum because dx is directly defined to be multiplying the entire series. But I have failed to so.
I am really confused by this. But I agree that dy/dx can be treated as a fraction
eg:
$ dy = f*dg + g*df $
$ = dy/dx = f*dg/dx + g*df/dx $
$ = y' = f*g' + g*f' $
Thus proving product rule.
but not in this context.
Because the "dx" can be easily any other symbol, but as long as it represents the variable "x" of F(x)+c.
this thing has been giving me a headache for a while now.
