Question

$\underset{1}{\overset{4}{\int }}{\log }_e[x]dx$ equals ( [.] denotes greatest integer function)

• A. ${\log }_{e}6$
• B. ${\log }_{e}3$
• C. ${\log }_{e}2$
• D. ${\log }_{e}4$

Answer 1

Answer: (A)

$\underset{1}{\overset{4}{\int }}\log [x]dx=\underset{1}{\overset{2}{\int }}\log [x]dx+\underset{2}{\overset{3}{\int }}\log [x]dx+\underset{3}{\overset{4}{\int }}\log [x]dx$
$=\underset{1}{\overset{2}{\int }}\log 1dx+\underset{2}{\overset{3}{\int }}\log 2dx+\underset{3}{\overset{4}{\int }}\log 3dy$
$=\log 2+\log 3=\log 6$