Question

${\int }_0^{\infty }\left[2e^{-x}\right]dx$, where $[.]$ denotes the greatest integer function, is equal to

• A. 0
• B. $\ell n2$
• C. $e^2$
• D. $2e^{-1}$

Answer 1

The correct option is B $\ell n2$

The function $y=2e^{-x}$ is a monotonically decreasing function

At $x=0,y=2$

At $x=\ln 2,y=1$

Hence for all $x>\ln 2,y<1$

$\Rightarrow \left[y\right]=0$

So the integral,

$I={\int }_0^{\infty }\left[y\right]dx$

$={\int }_0^{\ln 2}dx$

$\therefore I=\ln 2$