The value of $\int_{-\pi}^{\pi} [\sin(1-x)dx]$ is equal to (where $[.]$ denotes the greatest integer function)
Step-by-Step Solution
Key Concept: The absolute value function behaves differently on integer and non-integer domains, requiring careful evaluation of the piecewise definition
We know that $|x| + |-x| = \begin{cases} -1, & x \in \mathbb{Z} \\ 0, & x \notin \mathbb{Z} \end{cases}$. So $\int_{-\pi}^\pi (|\sin t| + |-\sin t|)dt = 6\left(\int_0^1 (1)dt + \int_1^2 (1)dt + \int_2^3 (1)dt + \int_3^\pi (-1)dt\right) = 6((1-1)(\frac{2}{3} + \frac{2}{3} + \frac{2}{3})) = -12$.
Correct Answer: -12