Question

The value of integral ${\int }_1^{e^6}\left[\frac{logx}{3}\right]dx$, where [.] denotes the greatest integer function, is

• A. $0$
• B. $e^6-e^3$
• C. $e^6+e^3$
• D. $e^3-e^6$

Answer 1

The correct option is B $e^6-e^3$

$When1<x<e^3,\left[\frac{logx}{3}\right]=0$
$Andwhen{e}^3<x<e^6,\left[\frac{logx}{3}\right]=1$
$\therefore {\int }_1^{e^6}\left[\frac{logx}{3}\right]dx={\int }_1^{e^3}\left[\frac{logx}{3}\right]dx+{\int }_{e^3}^{e^6}\left[\frac{logx}{3}\right]dx$
$={\int }_1^{e^3}0dx+{\int }_{e^3}^{e^6}1dx=(e^6-e^3)$