Mathematics - Definite Integration Question with Solution | TestHub

$I={\int }_0^{2}x[2x]dx$

$$\begin{array}{rl}& ={\int }_0^{1/2}0\cdot dx+{\int }_{1/2}^{1}xdx+{\int }_1^{3/2}2xdx+{\int }_{3/2}^{2}3xdx\\ & =0+{\left[\frac{x^2}{2}\right]}_{1/2}^1+{\left[x^2\right]}_1^{3/2}+{\left[\frac{3x^2}{2}\right]}_{3/2}^2\end{array}$$

$$\begin{array}{rl}& =\frac{1}{2}\left(1-\frac{1}{4}\right)+\left(\frac{9}{4}-1\right)+\frac{3}{2}\left(4-\frac{9}{4}\right)\\ & =\frac{3}{8}+\frac{5}{4}+\frac{21}{8}\\ & =\frac{34}{8}=\frac{17}{4}\\ & \therefore \frac{4}{17}I=1\end{array}$$