_ ,0 ^ ,1000 e^ x - [x] dx is - Vidyadip

MCQ

$\int_{\,0}^{\,1000} {{e^{x - [x]}}dx} $ is

A
${e^{1000}} - 1$
B
$\frac{{{e^{1000}} - 1}}{{e - 1}}$
✓
$1000(e - 1)$
D
$\frac{{e - 1}}{{1000}}$

✓

Answer

Correct option: C.

$1000(e - 1)$

c
(c) ${e^{x - [x]}}$ is a periodic function with period $1$.

$\therefore \int_0^{1000} {{e^{x - [x]}}dx = 1000\int_0^1 {{e^{x - [x]}}dx} } $,

$[\because [x]=0,$ if $\,0 < x < 1]$

$ = 1000{\rm{ }}\,[{e^x}]_0^1$

$ = 1000\,(e - 1)$.

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