question_answer

Evaluate \[\int_{1}^{2}{{{\log }_{e}}[x]dx,}\] where \[\left[ . \right]\] denotes the greatest integer function.

Answer:

Let \[l=\int_{1}^{2}{{{\log }_{e}}[x]dx,}\] \[=\int_{1}^{2}{{{\log }_{e}}1dx,}\]    \[[\because \,\,[x]=1,\,\,x\in [1,\,\,2]]\] = 0 \[[\because \,\,{{\log }_{e}}1=0]\]