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) \[\frac{1}{2}\log (a\,b)\log \,\left( \frac{a}{b} \right)\]
Correct Answer: C
Solution :
Let \[I=\int_{a}^{b}{\frac{1}{x}\log x\,dx=(\log x\log x)_{a}^{b}}\]\[-\int_{a}^{b}{\frac{1}{x}\log x\,dx}\] \[\Rightarrow 2I=[{{(\log x)}^{2}}]_{a}^{b}\Rightarrow I=\frac{1}{2}[{{(\log b)}^{2}}-{{(\log a)}^{2}}]\] \[=\frac{1}{2}[(\log b+\log a)(\log b-\log a)]\]=\[\frac{1}{2}\log (ab)\log \left( \frac{b}{a} \right)\].
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