Question

$\int _ { a } ^ { b } \frac { \log x } { x } d x =$

• A. $\log \left( \frac { \log b } { \log a } \right)$
• B. $\log ( a b ) \log \left( \frac { b } { a } \right)$
• C. $\frac { 1 } { 2 } \log ( a b ) \log \left( \frac { b } { a } \right)$
• D. $\frac { 1 } { 2 } \log ( a b ) \log \left( \frac { a } { b } \right)$

Answer 1

Answer: (C)

$\frac { 1 } { 2 } \log ( a b ) \log \left( \frac { b } { a } \right)$

Answer 2

Let $I = \int _ { a } ^ { b } \frac { 1 } { x } \log x d x = ( \log x \log x ) _ { a } ^ { b }$ $- \int _ { a } ^ { b } \frac { 1 } { x } \log x d x$

$\Rightarrow 2 I = \left[ ( \log x ) ^ { 2 } \right] _ { a } ^ { b } \Rightarrow I = \frac { 1 } { 2 } \left[ ( \log b ) ^ { 2 } - ( \log a ) ^ { 2 } \right]$

$= \frac { 1 } { 2 } [ ( \log b + \log a ) ( \log b - \log a ) ]$=$\frac { 1 } { 2 } \log ( a b ) \log \left( \frac { b } { a } \right)$.