Question:
Evaluate:
$$
\int_{-\pi/2}^{\pi/2} \sin \left(x - [x]\right) dx
$$
where $[x]$ is the greatest integer function.
Evaluate:
$$
\int_{-\pi/2}^{\pi/2} \sin \left(x - [x]\right) dx
$$
where $[x]$ is the greatest integer function.
Show Hint
Break integral into intervals where greatest integer function is constant.
Updated On: Jun 4, 2025
- 0
- \(3(1 - \cos 1) + \sin 2 - \sin 1\)
- \(3(1 - \cos 1) + \cos 2 - \sin 1\)
- \( \cos 2 - \sin 2 \)
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The Correct Option is B
Solution and Explanation
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: \[ 3(1 - \cos 1) + \sin 2 - \sin 1 \]
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