Question

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

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

Answer 1

Answer: (A)

Given curves are $y=x{e}^x$ and $y=x{e}^{-x}$
Line $x=1$ meets the curves at $A\left(1,e\right)$ and $B\left(1,\frac{1}{e}\right)$.
Both the curves pass through origin.
$\therefore$ Required area $=\underset{0}{\overset{1}{\int }}\left(x{e}^x-x{e}^{-x}\right)dx$
$=\underset{0}{\overset{1}{\int }}x\left(e^x-e^{-x}\right)dx$
=$x\left(e^x-e^{-x}\right){|}_0^1-\underset{0}{\overset{1}{\int }}\left(e^x-e^{-x}\right)dx$
$=\left(e+\frac{1}{e}\right)-e^x{\left|{}_0^1+e^{-x}\right|}_0^1$
$=e+\frac{1}{e}-e+1+\frac{1}{e}-1=\frac{2}{e}$