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Question
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
Solution
\[\text{We have}, \]
\[I = \int\limits_0^1 \left\{ x \right\} dx\]
\[\text{We know} \left\{ x \right\} = x, 0 < x < 1\]
\[ \therefore I = \int\limits_0^1 x\ dx\]
\[ = \left[ \frac{x^2}{2} \right]_0^1 \]
\[ = \frac{1}{2} - \frac{0}{2}\]
\[ = \frac{1}{2}\]
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