Question

Evaluate: $$ \int_{-\pi/2}^{\pi/2} \sin \left(x - [x]\right) dx $$ where $[x]$ is the greatest integer function.

• A. 0
• B. \(3(1 - \cos 1) + \sin 2 - \sin 1\)
• C. \(3(1 - \cos 1) + \cos 2 - \sin 1\)
• D. \( \cos 2 - \sin 2 \)

Hint:

Break integral into intervals where greatest integer function is constant.

Answer 1

The Correct Option is B

Note the function is periodic with period 1 on intervals of length 1. Split integral over intervals where \( [x] \) is constant: \[ \int_{-2}^{-1} + \int_{-1}^{0} + \int_0^1 + \int_1^2 + \ldots \] Each integral evaluates to a function of sine and cosine. Sum all integrals and simplify to get the final answer: \[ 3(1 - \cos 1) + \sin 2 - \sin 1 \]