Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (A)

The graph of the curve $y={\log }_e(x+e)$ is as shown in the fig.

Required area $A=\underset{1-e}{\overset{0}{\int }}ydx=\underset{1-e}{\overset{0}{\int }}{\log }_e\left(x+e\right)dx$
put $x+e=t\Rightarrow dx=dt$ also At $x=1-e,t=1$
At $x=0,t=e$ $\therefore A=\underset{1}{\overset{e}{\int }}{\log }_{e}tdt=[t{\log }_{e}t-t{]}_1^e$
e - e - 0 +1 = 1
Hence the required area is 1 square unit.