Question

Evaluate \( \int_{-2}^{1} \left[ x + 1 \right] \, dx \), where \( [x] \) is the greatest integer function not greater than \( x \)

• A. 1
• B. 0
• C. -1
• D. 2

Hint:

When integrating with a greatest integer function, break the integral into intervals where the function is constant.

Answer 1

The Correct Option is B

Step 1: Break the integral into intervals based on the greatest integer function.
The greatest integer function \( \left[ x + 1 \right] \) takes integer values between the intervals where \( x \) is between integer values. Break the integral into separate intervals: - From \( -2 \) to \( -1 \), \( \left[ x + 1 \right] = -1 \) - From \( -1 \) to \( 0 \), \( \left[ x + 1 \right] = 0 \) - From \( 0 \) to \( 1 \), \( \left[ x + 1 \right] = 1 \) Step 2: Calculate the integral.
Now calculate each part of the integral: \[ \int_{-2}^{-1} (-1) dx = -1, \quad \int_{-1}^{0} (0) dx = 0, \quad \int_{0}^{1} (1) dx = 1 \] So the total integral is: \[ -1 + 0 + 1 = 0 \] Step 3: Conclusion.
The correct answer is (B) 0.