Question

The value of $\underset{1}{\overset{3}{\int }}{e}^{\left\{x\right\}}dx$ is equal to
(where $\{.\}$ denotes fractional part function)

• A. $e-1$
• B. $2(e-1)$
• C. $3(e-1)$
• D. $4(e-1)$

Answer 1

The correct option is B $2(e-1)$

$\begin{array}{cc}I& =\underset{1}{\overset{3}{\int }}{e}^{\left\{x\right\}}dx\\ & =\underset{1}{\overset{2}{\int }}{e}^{x-1}dx+\underset{2}{\overset{3}{\int }}{e}^{x-2}dx\\ & =\frac{1}{e}{e^x|}_1^2+{\frac{1}{e^2}{e}^x|}_2^3\\ & =\frac{1}{e}(e^2-e)+\frac{e^3-e^2}{e^2}\\ & =e-1+e-1=2(e-1)\end{array}$