Question

If \( f(x) = x - \lfloor x \rfloor \), for every real number \( x \), where \( \lfloor x \rfloor \) is the integral part of \( x \), then \[ \int f(x) \, dx \] is equal to

• A. \( 0 \)
• B. \( \frac{1}{2} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{2} \)

Hint:

The fractional part function is periodic, and its integral over one period is 0.

Answer 1

The Correct Option is A

Step 1: Understanding the function.
The function \( f(x) = x - \lfloor x \rfloor \) is the fractional part of \( x \), which is periodic with a period of 1. The integral over one period results in 0.

Step 2: Conclusion.
The value of the integral is 0, corresponding to option (1).