Question

${\int }_0^1\left\{x\right\}[x+1]dx=$[where $\left\{x\right\}$ denotes the fractional part of $x$ and $[.]$ denotes the greatest integer of x]

• A. 0
• B. $\frac{1}{2}$
• C. 1
• D. $\frac{1}{3}$

Answer 1

Answer: (B)

$\frac{1}{2}$

$I={\int }_0^1\left[x\right][x+1]dx={\int }_0^1(x-\left[x\right])[x+1]dx$
For $x\in \{0,1\}\left[x\right]=0,[x+1]=1$
$\therefore I={\int }_0^{1}xdx={\left(\frac{x^2}{2}\right)}_0^1=\frac{1}{2}$