Let [ · ] denote the greatest integer function. If 0 e 3 [ 1 e x – 1 ] d x = – log e 2 , then 3 is equal to __________. [2025]

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Solution

Q.
Let $[\cdot]$ denote the greatest integer function. If $\int_{0}^{e^{3}} [\frac{1}{e^{x - 1}}] d x = α - \log_{e} 2$ , then $α^{3}$ is equal to __________. [2025]

Ans.
(8)

Let $f (x) = \frac{1}{e^{x - 1}} = e^{1 - x}$

$f (0) = e^{1} = 2.71; f (e^{3}) = e^{1 - e^{3}} \in (0, 1)$

Let $f (x) = 2 \Rightarrow \frac{1}{e^{x - 1}} = 2 \Rightarrow x = 1 - \log_{e} 2$

and $f (x) = 1 \Rightarrow x = 1$

$∴ I = \int_{0}^{1 - \log_{e} 2} 2 d x + \int_{1 - \log_{e} 2}^{1} 1 d x + \int_{1}^{e^{3}} 0 d x$

$= 2 (1 - \log_{e} 2 - 0) + (1 - 1 + \log_{e} 2) + 0 = 2 - \log_{e} 2$

$∴$ $α = 2$

Hence, $α^{3} = 8$ .

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