Let [.] denote the greatest integer function. If $\int_0^{e^3}\left[\frac{1}{\mathrm{e}^{\mathrm{x}-1}}\right] \mathrm{dx}=\alpha-\log _{\mathrm{e}} 2$, then $\alpha^3$ is equal to _______ .

Question

Aicharya · Accepted Answer

$f(x)=\frac{1}{e^{x-1}}=e^{1-x}$
\begin{array}{c|c}\mathrm{f}(\mathrm{x})=2 & \mathrm{f}(\mathrm{x})=1 \\\frac{1}{\mathrm{e}^{\mathrm{x}-1}}=2 & \mathrm{x}=1 \\\mathrm{x}=1-\ln 2 &\end{array}
$f(0)=e^1=2.71$
$f\left(e^3\right)=e^{1-e^3} \in(0,1)$
$\mathrm{I}=\int_0^{1-\ell \mathrm{n} 2} 2 \mathrm{dx}+\int_{1-\ell \mathrm{n} 2}^1 1 \mathrm{dx}+\int_1^{\mathrm{e}^3} 0 \mathrm{dx}$
$=2(1-\ell \operatorname{n} 2-0)+1(1-1+\ln 2)+0$
$\alpha-\ln 2=2-\ln 2$
$\alpha=2$
$\alpha^3=8$

Let [.] denote the greatest integer function. If…

Solution