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!
To evaluate the integral ∫ |f(x)|dx, we can break it down into different cases based on the function f(x). In the case of ∫ |x - 2|dx, we have two different cases to consider: when (x - 2) is less than 0 and when (x - 2) is greater than or equal to 0.
Let's walk through the steps to evaluate the integral without using a calculator:
Case 1: (x - 2) < 0
In this case, |x - 2| simplifies to -(x - 2).
∫ |x - 2|dx = ∫ -(x - 2)dx
Integrating this expression gives:
= -∫ (x - 2)dx
= -[x^2/2 - 2x] + C
Case 2: (x - 2) >= 0
In this case, |x - 2| simplifies to (x - 2).
∫ |x - 2|dx = ∫ (x - 2)dx
Integrating this expression gives:
= ∫ (x - 2)dx
= [x^2/2 - 2x] + C
Combining the results from both cases, we get:
∫ |x - 2|dx = -[x^2/2 - 2x] + C for (x - 2) < 0
= [x^2/2 - 2x] + C for (x - 2) >= 0
To address the specific expression you mentioned:
∫ |x - 2|dx = 2 - [(x - 2)^2 * sgn(2 - x)]/2
It seems that the expression you mentioned is simplifying the result further by substituting sgn(2 - x) = (2 - x)/|2 - x|. This substitution is valid when (x - 2) < 0, as explained in Case 1.
It's important to note that the exact form of an integral may vary depending on how it is simplified and what assumptions are made. Different mathematicians may approach the simplification in slightly different ways while still getting a correct result.
I hope this explanation helps you understand how to evaluate the integral |x - 2|dx without using a calculator!