Question

The value of integral $\underset{1}{\overset{e^3}{\int }}\left[\ln x\right]dx$ is
(where $[.]$ denotes the greatest integer function)

• A. $2e^3-e^2-e$
• B. $2e^3+e^2+e$
• C. $-2e^3-e^2-e$
• D. $2e^3-e^2+e$

Answer 1

The correct option is A $2e^3-e^2-e$

Let $I=\underset{1}{\overset{e^3}{\int }}\left[\ln x\right]dx$
$=\underset{1}{\overset{e}{\int }}0dx+\underset{e}{\overset{e^2}{\int }}1dx+\underset{e^2}{\overset{e^3}{\int }}2dx$$=0+e^2-e+2(e^3-e^2)$
$=2e^3-e^2-e$