Int Dx =/= X+c

[mishin05]

0--------[math]x[/math]----------A-----------[math]C[/math]------------B>[math]t[/math]

 

0->[math]t[/math] - Axis of abscisses,

 

{A} - Mobile point, therefore [math]x=|OA|[/math] - a variable,

 

[math]|AB|=C=const.[/math]

 

' 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:

 

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

 

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

 

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

 

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

 

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

 

[math]\displaystyle\int dx\not=x+C[/math], because [math]\displaystyle\int dx=|OA|[/math], [math]\displaystyle x+C=|OB|[/math].

 

The uncertain integral is limited by an integration variable. The certain integral is limited by two values of a variable of integration.

 

Geometrical interpretation of formula [math]\displaystyle U\cdot V=\int UdV+\int VdU[/math] for the elementary functions shows that both uncertain integrals are limited by arguments [math]V[/math] and [math]U[/math], therefore sum [math]\displaystyle\int UdV+\int VdU[/math] is equal to the area of rectangle [math]\displaystyle U\cdot V.[/math]

 

Error of an official science that on the basis: [math] \frac {dt} {dx} = \frac {dx} {dx} =1 [/math] the conclusion that [math] x=t [/math], because actually [math] t=x+C [/math] has been drawn.

It only one of many errors of an official science. I develop "the Structural analysis" which cleans all errors!