The value of $\int_{-1}^{1} |x - [x]| dx$ (where $[.]$ denotes greatest integer function) is

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Aicharya · Accepted Answer

$\int_{-1}^{1}(x-[x])dx = \int_{-1}^{1}xdx - \int_{-1}^{1}[x]dx$ $= \left[\frac{x^{2}}{2}ight]_{-1}^{1} - \left[\int_{-1}^{0}[x]dx + \int_{0}^{1}[x]dxight]$ $= $\frac{1}{2}$[1 - (-1)] - \left[\int_{-1}^{0}(-1)dx + \int_{0}^{1}0dxight]$ $\begin{aligned} &	ext{If } -1 \leq x < 0, [x] = -1\ &	ext{If } 0 \leq x < 1, [x] = 0 \end{aligned}$ $= 0 - [-x]_{-1}^{0} - 0 = 0 - [-0 - (-1)] = 1$

The value of $\int_{-1}^{1} |x - [x]| dx$ (where $[.]$ denotes greatest integer function) is

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