Question

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

• A. $0$
• B. $ln2$
• C. $e^2$
• D. $2e^{-1}$

Answer 1

The correct option is B $ln2$

The function $y=2e^{-x}$ is a monotonically decreasing function
at $x=0,y=2$
at $x=\ln 2,y=1$
Hence $\forall x>\ln \left(2\right)$
$y<1=>\left[y\right]=0$
So the integral $I={\int }_0^{\infty }\left[y\right]dx$
$I={\int }_0^{\ln 2}1dx$
$I=\ln 2$