Question

The area of the region bounded by the curves $y=ex\log x$ and $y=\frac{\log x}{ex}$ is:

• A. $\frac{e^2-5}{4e}$
• B. $e-\frac{5}{4}$
• C. $\frac{e}{4}-5$
• D. $\frac{e}{4}-\frac{1}{4e}$

Answer 1

The correct option is A $\frac{e^2-5}{4e}$

$|\stackrel{1}{\underset{1/e}{\int }}ex\log x-\frac{\log x}{ex}dx|$
$|e\stackrel{1}{\underset{1/e}{\int }}x\log x-\frac{1}{e}\stackrel{1}{\underset{1/e}{\int }}\frac{\log x}{x}dx|$
$\left|e(logx\frac{x^2}{2}-\frac{1}{4}{x}^2){|{}_{1/e}^1-\frac{1}{e}\frac{(\log x{)}^2}{2}|}_{1/e}^1\right|$
$\left|e(\frac{-1}{4}-(\frac{(-1)e^{-2}}{2}-\frac{1}{4}{e}^{-2})-1/e(0-\left(\frac{1}{2}\right)))\right|$
$\frac{-e}{4}+\left(\frac{3}{4}{e}^{-1}\right)+\frac{2}{4e}$
$|\frac{-e}{4}+\frac{5}{4e}|=\frac{e^2-5}{4e}squnits.$