Let $[\cdot]$ denote the greatest integer function. Then $\displaystyle\int_{-\pi/2}^{\pi/2}\left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right)dx$ is equal to:
Step-by-Step Solution
Key Concept: Split into intervals $[-\pi/2,-1)$, $[-1,0)$, $[0,1)$, $[1,\pi/2]$ where $[\sin x]$ and $[\cos x]$ are constant on each piece.
$I=11\pi+2$.
Correct Answer: 3