Question

${\int }_0^{2}x\left[\frac{x^2+4}{2}\right]dx=$ where [.] or the greatest integer function

• A. $4$
• B. $5$
• C. $3$
• D. $6$

Answer 1

The correct option is B $5$

${\int }_0^{2}x\left[\frac{x^2+4}{2}\right]dx$ $={\int }_0^{2}x[\frac{x^2}{2}+2]dx$
${\int }_0^{2}x\left\{\right[\frac{x^2}{2}]+2\}dx$
$={\int }_0^{\sqrt{2}}x.0dx+{\int }_{\sqrt{2}}^{2}x.1dx+4$ ....$(\because xϵ[0,\sqrt{2})\Rightarrow \left[\frac{x^2}{2}\right]=0$ and $xϵ[\sqrt{2},2)\Rightarrow \left[\frac{x^2}{2}\right]=1$
${\left(\frac{x^2}{1}\right)}_{\sqrt{2}}^2+4=(2-1)+4=5$
Ans: B