question_answer

The area of the figure bounded by \[y=\ell nx\] and  \[y={{(\ell nx)}^{2}}\]is -

A) \[3-e\]

B) \[3+e\]

C) \[5-e\]

D) \[5+e\]

Correct Answer: A

Solution :

Required Area \[=\int\limits_{1}^{e}{\left( \ell nx-{{(\ell nx)}^{2}} \right)\,\,dx}\]

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

\[=\int\limits_{0}^{1}{(t-{{t}^{2}}){{e}^{t}}dt={{e}^{t}}\left[ \left( t-{{t}^{2}} \right)-(1-2t)+(-\,2) \right]_{0}^{1}}\]

\[=3-e\]