Evaluating, Integral |f(x)|dx?

[GreenPrint]

Homework Statement

ok so apparently there's no way to express
integral |f(x)|dx
in standard mathematical functions... which I don't exactly buy...
but you the issue came up when I was trying to evaluate
integral |x - 2|dx
and apparently this is correct
integral |x-2|dx = 2 - [(x-2)^2sgn(2-x)]/2
now I'm not saying that it's wrong or anything I'm just carious as to why it's correct and if somebody could show me how one would get to that without a calculator and done by hand somehow... any help would be great... if you don't know the signum function, sgn(x), is defined as sgn(x) = x/|x| = e^(i arg(x)) = x/SQRT(x^2)
were arg(x) is the complex argument function

THANKS!

Homework Equations

The Attempt at a Solution

 

[GreenPrint]

well I understand what this term is
[(x-2)^2sgn(2-x)]/2
but I don't why we do 2 - this term?

 

[GreenPrint]

I'm also a tad bit confused as why it's
sgn(2-x) and not sgn(x-2)?

 

[Dick]

well I understand what this term is
[(x-2)^2sgn(2-x)]/2
but I don't why we do 2 - this term?

The '2' is irrelevant. It's part of the '+C' when you integrate. You can change it to '3' if you want. It doesn't make any difference. It's sgn(2-x) because they also arbitrarily put a '-' in front of the (x-2)^2/2. Try working out the integral of |t| from 0 to x, using that |t|=t if t>0 and |t|=(-t) if t<0. If you get that then just change t to t-2. It's a strange way to express the answer in several ways.