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In the notation here, the integrand denotes the function obtained by differentiating the antiderivative; the differential indicates the variable you differentiate the antiderivative with respect to, to get the integrand.
In the equation
we can interpret dx as an infinitesimal change in the variable x and dA(x) as the corresponding change in A(x).
With this interpretation we can write
dA = f(x)dx (*)
and applying the integral sign to each sides,
Thus our notation corresponds to the claim that for any antiderivative A we have,
.