A) \[2/3\]
B) \[1/3\]
C) \[3/2\]
D) \[\ln 2\]
Correct Answer: A
Solution :
\[I=\int_{1}^{{{e}^{2}}}{\frac{dx}{x{{(1+\ln x)}^{2}}}}\] Let \[(1+\ln x)=t\] Þ \[dt=\frac{1}{x}dx\] Now, when \[x=1\to {{e}^{2}}\], then \[t=1\to 3\] \[\therefore \] \[I=\int_{1}^{3}{\frac{dt}{{{t}^{2}}}=\left[ \frac{-1}{t} \right]_{1}^{3}=-\left[ \frac{1}{3}-1 \right]}=\frac{2}{3}\].
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