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JEE Mains · Maths · STD 12 - 8. Application and integration
If the area of the bounded region \(R=\left\{(x, y): \max \left\{0, \log _{e} x\right\} \leq y \leq 2^{x}, \frac{1}{2} \leq x \leq 2\right\}\) is, \(\alpha\left(\log _{e} 2\right)^{-1}+\beta\left(\log _{e} 2\right)+\gamma\), then the value of \((\alpha+\beta-2 \gamma)^{2}\) is equal to:
- A \(4\)
- B \(1\)
- C \(8\)
- D \(2\)
Answer & Solution
Correct Answer
(D) \(2\)
Step-by-step Solution
Detailed explanation
\(\mathrm{R}\left\{(x, y): \max \left(0, \log _{e} x\right) \leq y \leq 2^{x}, \frac{1}{2} \leq x \leq 2\right\}\) \(\int_{\frac{1}{2}}^{2} 2^{x} \,d x-\int_{1}^{2} \ln x\, d x\) \(\Rightarrow\left[\frac{2^{x}}{\ln 2}\right]_{1 / 2}^{2}-[x \ln x-x]_{1}^{2}\)…
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