question_answer

The area of the figure bounded by \[y=In\,\,x\] and \[y={{(Inx)}^{2}}\] is

A) \[3-e\]               

B) \[3+e\]

C) \[5-e\]               

D) \[5+e\]

Correct Answer: A

Solution :

Required area \[=\int\limits_{1}^{e}{(\ell n\,\,x-{{(\ell n\ x)}^{2}})}\,\,dx\]

Put \[\ell n\,\,x=t\]          \[dx={{e}^{t}}\,dt\]

\[\Rightarrow \]   \[=\int\limits_{0}^{1}{(t-{{t}^{2}})}\,{{e}^{t}}dt\]

\[\Rightarrow \]   \[{{e}^{t}}[(t-{{t}^{2}})-(1-2t)+(-\,2)]_{0}^{1}\]

\[\Rightarrow \]   \[3-e\]