A) \[\frac{1}{4}\log {{x}^{2}}\]
B) \[\frac{1}{4}{{(\log x)}^{2}}\]
C) \[\frac{1}{2}{{(\log x)}^{2}}\]
D) \[\log x\]
Correct Answer: C
Solution :
\[f\left( \frac{1}{r} \right)=\int\limits_{1}^{1/x}{\frac{\ln t}{1+t}dt}\] \[let\,t=\frac{1}{z}\] \[d\,t=-\frac{1}{{{z}^{2}}}dz\] \[=\int\limits_{1}^{x}{\frac{\ln z}{z\left( z+1 \right)}}dx\] \[t\left( x \right)+f\left( \frac{1}{r} \right)=\int\limits_{1}^{x}{\frac{\ln x}{z}}dz\] \[=\left[ \frac{{{\left( \ln z \right)}^{2}}}{2} \right]_{1}^{x}=\frac{{{\left( \ln x \right)}^{2}}}{2}\]
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