Question

The value of ${\int }_1^2(x^{\left[x^2\right]}+[x^2{]}^x)dx$, where $[.]$ denotes the greatest integer function, is equal to

• A. $\frac{5}{4}+\sqrt{3}+(2^{\sqrt{3}}-2^{\sqrt{2}})+\frac{1}{\log 3}(9-3^{\sqrt{3}})$
• B. $\frac{5}{4}+\sqrt{3}+\frac{\sqrt{2}}{3}+\frac{1}{\log 2}(2^{\sqrt{3}}-2^{\sqrt{2}})+\frac{1}{\log 3}(9-3^{\sqrt{3}})$
• C. $\frac{5}{4}+\frac{\sqrt{2}}{3}+\frac{1}{\log 2}(2^{\sqrt{3}}-2^{\sqrt{2}})+\frac{1}{\log 3}(9-3^{\sqrt{3}})$
• D. none of the above

Answer 1

Answer: (B)

$\frac{5}{4}+\sqrt{3}+\frac{\sqrt{2}}{3}+\frac{1}{\log 2}(2^{\sqrt{3}}-2^{\sqrt{2}})+\frac{1}{\log 3}(9-3^{\sqrt{3}})$

${\int }_1^2(x^{\left[x^2\right]}+{\left[x^2\right]}^x)dx={\int }_1^{\sqrt{2}}(x^{\left[x^2\right]}+{\left[x^2\right]}^x)dx+{\int }_{\sqrt{2}}^{\sqrt{3}}(x^{\left[x^2\right]}+{\left[x^2\right]}^x)dx+{\int }_{\sqrt{3}}^2(x^{\left[x^2\right]}+{\left[x^2\right]}^x)dx={\int }_1^{\sqrt{2}}(x+1)dx+\int (x^2+2^x)dx+{\int }_{\sqrt{3}}^2(x^3+3^x)dx$
$=\frac{5}{4}+\sqrt{3}+\frac{2}{\sqrt{3}}+\frac{1}{\log 2}(2^{\sqrt{3}}-2^{\sqrt{2}})+\frac{1}{\log 3}(9-3^{\sqrt{3}})$