A) \[\frac{1}{2}-e-\frac{1}{{{e}_{2}}}\]
B) \[-\frac{1}{2}+\frac{1}{e}-\frac{1}{2{{e}^{2}}}\]
C) \[\frac{3}{2}-\frac{1}{6}-\frac{1}{2{{e}^{2}}}\]
D) \[\frac{3}{2}-e-\frac{1}{2{{e}^{2}}}\]
Correct Answer: D
Solution :
| \[\int\limits_{1}^{e}{{{\left( \frac{x}{e} \right)}^{2x}}{{\log }_{e}}x.dx-\int\limits_{1}^{e}{\left( \frac{e}{x} \right){{\log }_{e}}x.dx}}\] |
| Let, \[{{\left( \frac{x}{e} \right)}^{2x}}=t,{{\left( \frac{e}{x} \right)}^{x}}=v\] |
| \[=\frac{1}{2}\int\limits_{\left( \frac{1}{e} \right)}^{1}{dt}+\int\limits_{e}^{1}{dv}\]\[=\frac{1}{2}\left( 1-\frac{1}{{{e}^{2}}} \right)+(1-e)=\frac{3}{2}-\frac{1}{2{{e}^{2}}}-e.\] |
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