Question

The value of ${\int }_5^{10}\left[\frac{x-5}{5}\right]dx$ where $[.]$ represents greatest integer function is

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is D $0$

$f\left(x\right)=x-5\in (0,5)\forall x\in (5,10)$

$g\left(x\right)=f\frac{\left(x\right)}{5}=\frac{x-5}{5}\in (0,1)\forall x\in (5,10)$

$\left[g\right(x\left)\right]=\left[\frac{x-5}{5}\right]=0$

$\therefore I={\int }_5^{10}\left[\frac{x-5}{5}\right]dx=0$

$\therefore$ Option D is correct