Question

The area bounded by the curves $y=\ln x$ and $y=(\ln x{)}^2$ is

• A. $(3-e)\text{sq. units}$
• B. $(3+e)\text{sq. units}$
• C. $3\text{sq. units}$
• D. $(3+2e)\text{sq. units}$

Answer 1

The correct option is A $(3-e)\text{sq. units}$

The bounded region is shown in the figure :


So, the required area is :
$=\underset{1}{\overset{e}{\int }}(\ln x-(\ln x{)}^2)dx$
Using ILATE rule in the integral $(\ln x{)}^2,$ we get
$=\underset{1}{\overset{e}{\int }}\ln xdx-{\left[x\right(\ln x{)}^2]}_1^e+\int 2\ln xdx$
$=3\underset{1}{\overset{e}{\int }}\ln xdx-{\left[x\right(\ln x{)}^2]}_1^e$
$=3[x\ln x-x{]}_1^e-[x(\ln x{)}^2{]}_1^e$
$=(3-e)\text{sq. units}$