MCQ
$\int\limits_0^\infty {} [2 e^{-x}]\, dx$ where $[x]$ denotes the greatest integer function is
- A$0$
- ✓$ln\, 2$
- C$e^2$
- D$2/e$
alternatively, put $e^{-x} = t ; - x = ln\, t ; dx = \frac{1}{t}\,dt$
$\int\limits_0^1 {[2t]\frac{1}{t}dt} $ $-\int\limits_0^1 {[2t]\frac{{dt}}{t}} $ ;
$\int\limits_0^{\frac{1}{2}} {0\,dt} $ $+\int\limits_{\frac{1}{2}}^{\frac{1}{2}} {\frac{{dt}}{t}} $
$=\left. {\ln \,t} \right]_{\frac{1}{2}}^{\,1}$
$= [1] - ln\, \frac{1}{2} = ln\, 2$
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