Question

The area of the region bounded by the curves $y={xe}^x,y={xe}^{-x}$ and the line $x=1$ is

• A. $\frac{4}{e}$
• B. $\frac{3}{e}$
• C. $\frac{2}{e}$
• D. $\frac{1}{e}$

Answer 1

The correct option is C $\frac{2}{e}$

$A={\int }_0^1(x{e}^x-x{e}^{-x})dx$

$={\int }_0^{1}x(e^x-e^{-x})dx$

$=(e^x+e^{-x}){\int }_0^1={\int }_0^1(e^x+e^{-x})dx$

$=(e+e^{-1})-[e^x-e^{-x}{]}_0^1$

$=e+e^{-1}-(e-e^{-1})$

$=2e^{-1}$

$A=\frac{2}{e}$ Option $C$ is correct