A) \[2\log \,\left( 3/2{{e}^{3}} \right)\]
B) \[\log (3/{{e}^{3}})\]
C) \[4\log (3/{{e}^{3}})\]
D) None of these
Correct Answer: A
Solution :
We have \[\int_{3}^{8}{\frac{2-3x}{x\sqrt{1+x}}dx=I}\] Put \[1+x={{t}^{2}}\Rightarrow dx=2t\,dt\] When \[x=3\to 8,\] then \[t=2\to 3\] \ \[I=2\int_{2}^{3}{\frac{5-3{{t}^{2}}}{{{t}^{2}}-1}dt}\]; \[I=2\int_{2}^{3}{\left( \frac{2}{{{t}^{2}}-1}-3 \right)}\,dt\] \[I=2\left[ \frac{2}{2.1}\log \frac{t-1}{t+1}-3t \right]_{2}^{3}\];\[I=2\log \left( \frac{3}{2{{e}^{3}}} \right)\].
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