Question

\[\int\limits_0^{15} \left[ x \right] dx .\]

Answer 1

\[\text{We have}, \]
\[I = \int\limits_0^{1 . 5} \left[ x \right] dx\]
\[ = \int_0^1 \left[ x \right] dx + \int_1^{1 . 5} \left[ x \right] dx\]
\[ = \int_0^1 \left( 0 \right) dx + \int_1^{1 . 5} \left( 1 \right)dx ................\left[\because \left[ x \right] = \begin{cases}0&& 0 \leq x < 1\\1&& 1 \leq x < 1 . 5\end{cases} \right]\]
\[ = 0 + \left[ x \right]_1^{1 . 5} \]
\[ = 1 . 5 - 1\]
\[ = 0 . 5\]
\[ = \frac{1}{2}\]