Question

${\int }_0^{\infty }\left[2e^{-x}\right]dx$ where [x] denotes the greatest interger function is

• A. $0$
• B. $ln2$
• C. $e^2$
• D. $2/e$

Answer 1

The correct option is B $ln2$

For $0<x<ln2$
$\left[{2e}^{-x}\right]=1$ otherwise $0$
$\Rightarrow {\int }_0^{\infty }\left[{2e}^{-x}\right]dx={\int }_0^{ln2}1.dx+{\int }_{ln2}^{\infty }0dx$

$={\left[x\right]}_0^{\ln 2}+0=ln2$