# Why does the dx in the integral have algebriac properties?

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.

• 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. – Lee David Chung Lin Apr 14 at 2:49
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