Question

${\int }_0^2\left(\right[x{]}^2-\left[x^2\right])dx$ is equal to (where $[.]$ denotes the greatest integer function)

• A. $4+2\sqrt{3}+\sqrt{2}$
• B. $-4+\sqrt{3}+\sqrt{2}$
• C. $4+\sqrt{3}-\sqrt{2}$
• D. $-2+\sqrt{3}+\sqrt{2}$

Answer 1

The correct option is B $-4+\sqrt{3}+\sqrt{2}$

${\int }_0^2\left(\right[x{]}^2-\left[x^2\right])dx$
${\int }_0^2[x{]}^{2}dx-{\int }_0^2[x^2]dx$
$({\int }_0^{1}0dx+{\int }_1^{2}1dx)-({\int }_0^{1}0dx+{\int }_1^{\sqrt{2}}1dx+{\int }_{\sqrt{2}}^{\sqrt{3}}2dx+{\int }_{\sqrt{3}}^{2}3dx)$
$(0+1)-[0+(\sqrt{2}-1)+2(\sqrt{3}-\sqrt{2})+3(2-\sqrt{3}\left)\right]$
$1-[\sqrt{2}-1+2\sqrt{3}-2\sqrt{2}+6-3\sqrt{3}]$
$1-[5-\sqrt{2}-\sqrt{3}]$
$=-4+\sqrt{3}+\sqrt{2}$