The Structural Analysis

[mishin05]

The classical analysis has errors. That to eliminate them I have thought up the structural analysis.

 

The basic formula:[math]\displaystyle \int f(a+x)dx=\int\limits_{a}^{a+x} f(t)dt[/math].

 

It leads to following contradictions for example:

 

the Structural analysis:

 

[math]\displaystyle\int(a+x)dx=\frac{(a+x)^2}{2}-\frac{a^2}{2}\not=\int(a+x)d(a+x)[/math];

 

[math]\displaystyle\int\limits_{0}^{x}2tdt=\int2xdx=x^2[/math];

 

[math]\displaystyle\int\limits_{0}^{\sqrt{x^{2}+C}}2tdt=x^2+C[/math].

 

The historical background: HERE

 

 

P.S.

 

1. [math]\displaystyle \int\limits_{a}^{a+x}tdt=\frac{(a+x)^2}{2}-\frac{a^2}{2}[/math];

 

2. [math]\displaystyle \int (a+x)dx=(a+x)x-\int xd(a+x)[/math];

 

3. [math]\displaystyle \int (a+x)dx=(a+x)x-\int xda-\int xdx[/math];

 

4. [math]\displaystyle \int (a+x)dx=(a+x)x-\int xdx[/math];

 

5. [math]\displaystyle \int (a+x)dx=\left(\frac{(a+x)^2}{2}-\frac{a^2}{2}+\frac{x^2}{2}\right)-\int xdx[/math];

 

6. [math]\displaystyle \int (a+x)dx=\left(\frac{(a+x)^2}{2}-\frac{a^2}{2}+\int xdx\right)-\int xdx[/math];

 

7. [math]\displaystyle \int (a+x)dx=\left(\frac{(a+x)^2}{2}-\frac{a^2}{2}\right)+\left(\int xdx-\int xdx\right)[/math];

 

8. [math]\displaystyle \int (a+x)dx=\frac{(a+x)^2}{2}-\frac{a^2}{2}[/math];

 

1. and 8. [math]\rightarrow[/math][math]\displaystyle \int f(a+x)dx=\int\limits_{a}^{a+x} f(t)dt[/math].

[Qfwfq]

The problem stems from the fact that you are talking about both indefinite and definite integrals, but you are leaving out the arbitrarity of additive constant in the former case. Strictly, your first equation should be:

 

[math]\displaystyle \int f(a+x)dx=\int\limits_{a}^{a+x} f(t)dt+c[/math]

 

A value of [imath]c[/imath] other than 0 is the same as changing the lower bound of the interval to some value other than [imath]a[/imath]. Forgetting the arbitrarity leads to the discrepancies you are discussing. In fact, when we are taught indefinite integrals in the usual manner, we are being taght a slight hash but it is just a first simple step before learning the real concept. For instance we write:

 

[math]\int x\,dx=\frac{x^2}{2}+c[/math] equivalent to [math]\frac{d}{dx}\frac{x^2}{2}=x[/math]

 

and it would be more precise to write:

 

[math]\int_a^x t\,dt=\frac{x^2}{2}-\frac{a^2}{2}[/math]

 

The simpler notation just gives a relation between functions (integrand and its primitives) without fussing over details.

[mishin05]

The problem stems from the fact that you are talking about both indefinite and definite integrals, but you are leaving out the arbitrarity of additive constant in the former case.

 

 

' the Structural analysis ' approves, that record of the mathematical analysis [math]\displaystyle\int dx=x+C[/math] makes no sense, since in it actually three are laid various integral of Reimann!

 

1. If [math]\displaystyle t=x+C[/math],

 

[math]\displaystyle\frac{dt}{dt} = \frac {d (x+C)}{d (x+C)}=1.[/math]

 

[math]\displaystyle\int \limits _{0}^{t}dt=t = \int\limits_{0}^{x+C}d(x+C) = \int\limits_{0}^{x+C}dt=x+C ====== \int d(x+C)=x+C.[/math]

 

2. Special case [math]\displaystyle t=x+C[/math] at [math]\displaystyle C=0 [/math]

 

[math]\displaystyle\frac{dt}{dt} = \frac{dx}{dx}=1.[/math]

 

[math]\displaystyle\int\limits_{0}^{t}dt[/math] [math] (t=x) [/math] [math]\displaystyle = \int \limits_{0}^{x}dx=x ================ \int dx=x.[/math]

 

3. [math]\displaystyle\frac{dt}{d(t-C)}=\frac{dt}{dx} = \frac{d (x+C)}{dx} = \frac {dx}{dx} =1.[/math]

 

[math]\displaystyle\int\limits_{0}^{t-C} dt=x ======================== \int \limits_{0}^{x}dt=x[/math].

[mishin05]

To 'Qfwfq'

All right, if don't want, that I here discussed the mathematician, prompt on what other site to me to leave, where there are more than people.

[Nootropic]

Not sure what you're trying to do with all those undefined terms and incorrect integrals, but Calculus (and it's rigorous counterpart, Analysis) has been around for quite some time and I assure you there is nothing wrong with integrating a constant! Perhaps for more interesting topics in integration, try reading up on the Riemann-Stieljes integral or the Lebesgue integral.