intlimits_a^b [x] ,dx + intlimits_a^b [-x] ,d | vedclass

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(C) We know that for the greatest integer function $[x]$,the sum $[x] + [-x]$ is defined as:$[x] + [-x] = \begin{cases} 0, & 	ext{if } x \in \mathbb{

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$a - b$

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(C) We know that for the greatest integer function $[x]$,the sum $[x] + [-x]$ is defined as:$[x] + [-x] = \begin{cases} 0, & 	ext{if } x \in \mathbb{Z} \ -1, & 	ext{if } x 
otin \mathbb{Z} \end{cases}$Since the set of integers $\mathbb{Z}$ has measure zero in the interval $[a, b]$,the integral of the function over the interval $[a, b]$ is equivalent to integrating the constant value $-1$ over the interval $[a, b]$.Therefore,$I = \int\limits_a^b ([x] + [-x]) \,dx = \int\limits_a^b (-1) \,dx$$I = -[x]_a^b = -(b - a) = a - b$.

$\int\limits_a^b [x] \,dx + \int\limits_a^b [-x] \,dx$,where $[.]$ denotes the greatest integer function,is equal to:

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