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Q. $\int\limits_1^4 \log _e[x] d x$ equals ( [.] denotes greatest integer function)

Integrals

Solution:

$\int\limits_1^4 \log [x] d x=\int\limits_1^2 \log [x] d x+\int\limits_2^3 \log [x] d x+\int\limits_3^4 \log [x] d x$
$=\int\limits_1^2 \log 1 d x+\int\limits_2^3 \log 2 d x+\int\limits_3^4 \log 3 d y$
$=\log 2+\log 3=\log 6$