Question

The value of ${\int }_0^{\left[x\right]}\left\{x\right\}dx$ (where [.] and {.} denotes greatest integer and fraction part function respectively ) is

• A. $\frac{1}{2}\left[x\right]$
• B. $2\left[x\right]$
• C. $\frac{1}{2\left[x\right]}$
• D. $\left[x\right]$

Answer 1

The correct option is A $\frac{1}{2}\left[x\right]$

${\int }_0^{\left[x\right]}\left\{x\right\}dx$

we know $\left\{x\right\}$ is a periodic function with period $1.$
As $\left[x\right]$ is an integer:

${\int }_0^{\left[x\right]}\left\{x\right\}dx=\left[x\right]{\int }_0^1\left\{x\right\}dx$
In $[0,1)$, $\left\{x\right\}=x$.

$\therefore {\int }_0^1\left\{x\right\}dx=\frac{1}{2}$
Therefore, required answer: $\frac{1}{2}\left[x\right]$