calculous in the curve space theorem 3

Cao's theorem 3
From when x→0 there are sin x=x, ex-1=x, ln(1+x)=x, (1+x)^а-1=аx, we
can conclude follow theorem
1, ∵sin dx=dx

∴ ∫sin dx dx=∫dxdx=1

2, ∵ edx-1=dx

∴ ∫(edx-1)dx=∫dxdx=1

3, ∵ ln(1+dx)=dx

∴ ∫ln(1+dx)dx=∫dxdx=1

4, ∵ (1+dx)^а-1=аdx

∴ ∫[(1+dx)^а-1]dx=∫аdxdx=а∫dxdx=а

These all can show even if a very tiny digital such as dx in the
integral formula, we cann't deal it with 0 and then calculate them
again, that is incorrect. Because even if a very tiny digital such as
dx→0 , as after we calculate the integral formula , it is a number
that cann't be ignored. The 4 can explain it throughly.

caoyan

2007-10-31
http://thre-firewh2.home.sunbo.net/index.php?xname=AB6BP01

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