_0^ [2 e^ -x ] , dx where [x] denotes the greatest integer functi… - Vidyadip

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
$2/e$

✓

Answer

Correct option: B.

$ln\, 2$

b
for $0 < x < ln\, 2,\, [2e^{-x}] = 1,$ otherwise zero $ \Rightarrow I = \int\limits_0^{\ell n\,2} {} \,dx + \int\limits_{\ell n\,2}^\infty {} 0 \,dx = ln\, 2$

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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