A) \[16\,\,\text{lo}{{\text{g}}_{e}}\,2\,\,-\,\frac{14}{3}\]
B) \[\text{lo}{{\text{g}}_{e}}\,2\,\,-\,\frac{7}{3}\]
C) \[8\,\,\text{lo}{{\text{g}}_{e}}\,2\,-\,\frac{14}{3}\]
D) \[8\,\,\text{lo}{{\text{g}}_{e}}\,2\,-\,6\]
Correct Answer: A
Solution :
\[xy\ge 8\,\text{and}\,1\le y\,\le {{x}^{2}}\] |
\[\text{A}=\int\limits_{1}^{2}{({{x}^{2}}-1)dx+\int\limits_{2}^{8}{\left( \frac{8}{x}\,-\,1 \right)\text{d}x}}\] |
\[{{\left. \text{A}=\frac{{{x}^{3}}}{3} \right|}^{2}}_{1}+8\,\text{In}\left. x \right|_{2}^{8}-1-6\] |
\[\text{A}\,=\,\left( \frac{8}{3}-\frac{1}{3} \right)+8(\text{In}8-\text{In}2)\,-\,7\] |
\[\text{A}=\frac{7}{3}-7+16\,\text{In}\,2\] |
\[\text{A}=16\,\text{In}\,2\,-\,\frac{14}{3}.\] |
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