Question

Let $A$ be the area enclosed by the curve $y=\ln (x+e),x=\ln \left(\frac{1}{y}\right)$ and line $y=0.$ Then the value of $\left[A\right],$ where $[.]$ is the greatest interger function, is

Answer 1

The given curves are
$y=\ln (x+e)x=\ln \left(\frac{1}{y}\right)\Rightarrow y=e^{-x}$
$y=0$
By using the transformation of $y=\ln x$ to $y=\ln (x+e)$
plot the graph of the equation
Required area
$=\underset{1-e}{\overset{0}{\int }}\ln (x+e)dx+\underset{0}{\overset{\infty }{\int }}{e}^{-x}dx=\underset{1}{\overset{e}{\int }}\ln xdx+\underset{0}{\overset{\infty }{\int }}{e}^{-x}dx={[x\ln x-x]}_1^e+{[-e^{-x}]}_0^{\infty }=2$