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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.

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  • $\begingroup$ Your question is kind of messy so I'm not sure what you really need, but please checkout this and that and some "Related" posts. $\endgroup$ – Lee David Chung Lin Apr 14 at 2:49
  • $\begingroup$ Hello and, again, welcome to the Aperture Science computer-aided enrichment center. $\endgroup$ – logo Apr 19 at 12:26

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