What is the integretion of modxdx?

Answer 1

mod x, or |x| is actually a conjunction of two functions:

1) x = -x, for x < 0

2) x = x, for x >= 0.

Whenever you're calculating integral of |x|, you have to consider those two functions, for example:

integral of |x| from -5 to 4 by dx is a sum of integrals of -x from -5 to 0 by dx and integral of x from 0 to 4 by dx.