Question

ok so apparently there's no way to express
integral |f(x)|dx
in standard mathematical functions... which I don't exactly buy...
but ya 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!

Answer 1

integral (abs(x-2)) dx

The issue here is that the abs ()changes sign when the argument is negative, so you have to counter act that with the signum function

Answer 2

http://en.wikipedia.org/wiki/Sign_function

Answer 3

ya i realized that but I still have no idea how you get 2 - [(x-2)^2sgn(2-x)]/2 by hand...

Answer 4

This has to be a definite integral, right?

otherswise,how did a constant get into the integration result?

I integrate it as (x-2)^2SGN(x-2)/2+C

if the limits are zero to x, then
int=(x-2)^2SGN(x-2)/2 -(-2)^2(SGN2)/2=
= 2-(x-2)^2SGN(2-x)/2

I hope this helps.