Question

if $[.]$ denotes the greatest integer function,then the value of ${\int }_1^2\left(\right[x^2]-[x^2\left]\right)dx$: is

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

Answer 1

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

${}_1^2\int \left(\right[x^2]-[x{]}^2)dx$

${=}_1^2\int \left[x^2\right]dx{-}_1^2\int [x{]}^{2}dx$

${=}_1^{\sqrt{2}}\int \left[x^2\right]dx{+}_{\sqrt{2}}^{\sqrt{3}}\left[x^2\right]dx{+}_{\sqrt{3}}^2\left[x^2\right]dx{-}_1^2\int [x{]}^{2}dx$

${=}_1^{\sqrt{2}}\int dx{+}_{\sqrt{2}}^{\sqrt{3}}2dx{+}_{\sqrt{3}}^2\int 3dx{-}_1^2\int dx$

$(\sqrt{2}-1)+2(\sqrt{3}-\sqrt{2})+3(2-\sqrt{3})-1$

$=\sqrt{2}-1+2\sqrt{3}-2\sqrt{2}+6-3\sqrt{3}-1$

$=4-\sqrt{3}-\sqrt{2}$