Question

$\underset{0}{\overset{3}{\int }}{e}^{\left[x\right]}dx$ equals to
(where $[.]$ denotes the greatest integer function)

• A. $1+e$
• B. $1+e+e^2$
• C. $1+e-e^2$
• D. $e^2+1$

Answer 1

The correct option is B $1+e+e^2$

$\begin{array}{cc}I& =\underset{0}{\overset{3}{\int }}{e}^{\left[x\right]}dx\\ & =\underset{0}{\overset{1}{\int }}{e}^{0}dx+\underset{1}{\overset{2}{\int }}{e}^{1}dx+\underset{2}{\overset{3}{\int }}{e}^{2}dx\\ & =1+e(2-1)+e^2(3-2)=1+e+e^2\end{array}$