Question

The area enclosed between the curve $y={\log }_e(x+e)$ and the coordinate axes is

Answer 1

Required Area $A=\underset{1-e}{\overset{0}{\int }}ydx=\underset{1-e}{\overset{0}{\int }}{\log }_e(x+e)dx$
Put $x+e=t\Rightarrow dx=dt$
also when $x=1-e,t=1$ and when $x=0,t=e$
$\therefore A=\underset{1}{\overset{e}{\int }}{\log }_{e}tdt={[t{\log }_{e}t-t]}_1^e$
$=e-e-0+1=1$ sq. unit