What is the integral of x mod x? Here mod is modulus of x.

The integral of the function \( x \mod x \) can be understood by first analyzing what \( x \mod x \) represents. The modulus operation gives the remainder of the division of \( x \) by \( x \). For any non-zero value of \( x \), this results in zero, since any number divided by itself leaves no remainder. However, at \( x = 0 \), the expression is undefined.

Function Behavior

Thus, we can summarize the behavior of the function:

• For \( x > 0 \): \( x \mod x = 0 \)
• For \( x < 0 \): \( x \mod x = 0 \)
• At \( x = 0 \): Undefined

Integral Calculation

Given that \( x \mod x = 0 \) for all \( x \) except at zero, the integral can be expressed as:

∫(x mod x) dx = ∫0 dx

This integral evaluates to:

C, where C is the constant of integration.

Final Result

Therefore, the integral of \( x \mod x \) is simply a constant:

∫(x mod x) dx = C