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JEE Main 2-April-2025 Paper - Shift 1 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 2-April-2025 Paper - Shift 14 Marks
Question
Let [•] denote the greatest integer function. If $\int_0^{ e ^3}\left[\frac{1}{ e ^{ x -1}}\right] dx =\alpha-\log _{ e } 2$, then $\alpha^3$ is equal to __________.
✓

Answer

(8)
Explanation: $f(x)=\frac{1}{e^{x-1}}=e^{1-x}$
$f(x)=2$ / $f(x)=1$
$\frac{1}{ e ^{ x -1}}=2$ / $x=1$
$x=1-\ln 2$
$f(0)=e^1=2.71$
$f\left(e^3\right)=e^{1-e^3} \in(0,1)$
$I =\int_0^{1-\ell n 2} 2 dx +\int_{1-\ell n 2}^1 1 dx +\int_1^{ e ^3} 0 dx$
$=2(1-\ell n 2-0)+1(1-1+\ell n 2)+0$
$\alpha-\ell n 2=2-\ell n 2$
$\alpha=2$
$\alpha^3=8$

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