JEE Mains · Maths · STD 12 - 7.2 definite integral
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 _______ .
- A 2
- B 4
- C 6
- D 8
Answer & Solution
Correct Answer
(D) 8
Step-by-step Solution
Detailed explanation
\(f(x)=\frac{1}{e^{x-1}}=e^{1-x}\) \(\begin{array}{c|c}\mathrm{f}(\mathrm{x})=2 & \mathrm{f}(\mathrm{x})=1 \\\frac{1}{\mathrm{e}^{\mathrm{x}-1}}=2 & \mathrm{x}=1 \\\mathrm{x}=1-\ln 2 &\end{array}\) \(f(0)=e^1=2.71\) \(f\left(e^3\right)=e^{1-e^3} \in(0,1)\)…
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