Question

$\underset{0}{\overset{2}{\int }}x\left[x\right]dx.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ I=\underset{0}{\overset{2}{\int }}x\left[x\right]dx \\ \mathrm{We}\mathrm{know}\mathrm{that}, \\ x\left[x\right]=\left\{\begin{array}{l}x\times 0,0<x<1\\ x\times 1,1<x<2\end{array}\right. \\ \mathrm{i}.\mathrm{e}., \\ x\left[x\right]=\left\{\begin{array}{l}0,0<x<1\\ x,1<x<2\end{array}\right. \\ \therefore I=\underset{0}{\overset{2}{\int }}x\left[x\right]dx \\ =\underset{0}{\overset{1}{\int }}x\left[x\right]dx+\underset{1}{\overset{2}{\int }}x\left[x\right]dx \\ =\underset{0}{\overset{1}{\int }}\left(0\right)dx+\underset{1}{\overset{2}{\int }}\left(x\right)dx \\ =0+{\left[\frac{x^2}{2}\right]}_1^2 \\ =\frac{2^2}{2}-\frac{1^2}{2} \\ =\frac{4}{2}-\frac{1}{2} \\ =\frac{3}{2} \end{gathered}$$