Question

\[\int_0^2 2x\left[ x \right]dx\]

Answer 1

\[\int_0^2 2x\left[ x \right]dx\]
\[ = \int_0^1 2x\left[ x \right]dx + \int_1^2 2x\left[ x \right]dx\]
\[ = \int_0^1 2x \times 0dx + \int_1^2 2x \times 1dx .................\left[ \left[ x \right] = \begin{cases}0, & 0 \leq x < 1 \\ 1, & 1 \leq x < 2\end{cases} \right]\]
\[ = 0 + 2 \int_1^2 xdx\]
\[ = \left.2 \times \frac{x^2}{2}\right|_1^2 \]
\[ = 4 - 1\]
\[ = 3\]