Question

The value of the definite integral $\int\limits_{-\infty}^{\ln 3}\left\{ e ^{ x }\right\} dx$ equals
[Note: $\{y\}$ denotes fractional part of $y$.]

• A. $3+\ln 2-2 \ln 3$
• B. $3-\ln 3$
• C. $2 \ln 3+\ln 2$
• D. 1

Answer 1

Answer: (A)

$I =\int\limits_{-\infty}^0 e ^{ x } dx +\int\limits_0^{\ln 2}\left( e ^{ x }-1\right) dx +\int\limits_{\ln 2}^{\ln 3}\left( e ^{ x }-2\right) dx =\left. e ^{ x }\right|_{-\infty} ^0+ e ^{ x }-\left. x \right|_0 ^{\ln 2}+ e ^{ x }-\left.2 x \right|_{\ln 2} ^{\ln 3}$
$=1+(2-\ln 2)-1+(3-2 \ln 3)-(2-2 \ln 2)=2-\ln 2+1-2 \ln 3+2 \ln 2$
$=3+\ln 2-2 \ln 3$