The area bounded by the curve $y=x{e}^{-x}; xy=0$ and $x=c,$ where $c$ is the $x$-coordinate of the curve's inflection point, is -

 

If $C_1\equiv y=\frac{1}{1+x^2}$ and $C_2\equiv y=\frac{x^2}{2}$ be two curves lying in $xy$ plane. Then 

$y=f\left(x\right)$  is a function which satisfies

$\left(\mathrm{i}\right) f\left(0\right)=0$ 

$\left(\mathrm{ii}\right) f^{\text{'}\text{'}}\left(x\right)=f^{\text{'}}\left(x\right)$ and

$\left(\mathrm{iii}\right) f^{\text{'}}\left(0\right)=1$.

Then, the area bounded by the graph of  $y=f\left(x\right),$ the lines $x=0,$ $x-1=0$ and $y+1=0,$ is