Question

The value of \[ \int_{-1}^{1} (x - \lfloor x \rfloor) dx \quad (\text{where} \, \lfloor \cdot \rfloor \, \text{denotes the greatest integer function}) \] is:

• A. 0
• B. 1
• C. 2
• D. None of these

Hint:

To evaluate integrals involving the greatest integer function, break the interval into subintervals where \( \lfloor x \rfloor \) is constant.

Answer 1

The Correct Option is A

Step 1: The greatest integer function, \( \lfloor x \rfloor \), takes the value of the greatest integer less than or equal to \( x \). The integral is split into two parts: \[ \int_{-1}^{0} (x - \lfloor x \rfloor) dx + \int_{0}^{1} (x - \lfloor x \rfloor) dx. \] Step 2: After evaluating, we find the result of the integral is 0.

Final Answer: \[ \boxed{0} \]