Question

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

• A. $\frac{5}{3}$
• B. $\frac{8}{3}$
• C. $\frac{3}{2}$
• D. $\frac{2}{3}$

Answer 1

The correct option is C $\frac{3}{2}$

We have,

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

Since,

$\left[x\right]=0,when0<x<1$

$\left[x\right]=1,when1<x<2$

Therefore

$I={\int }_0^{1}0dx+{\int }_1^{2}xdx$

$I=0+{\left[\frac{x^2}{2}\right]}_1^2$

$I=\frac{1}{2}(2^2-1^2)$

$I=\frac{1}{2}(4-1)$

$I=\frac{3}{2}$

Hence, Option $C$ is the answer.