What is the integral of mod x function?

The integral of the absolute value function, |x|, is given by the piecewise function ∫|x|dx = (x^2)/2 + C for x ≥ 0 and ∫|x|dx = (-x^2)/2 + C for x < 0, where C is the constant of integration. This is because the absolute value function changes its behavior at x = 0, resulting in two different expressions for the integral depending on the sign of x.