A) \[x\log (\log x)+\frac{x}{\log x}+c\]
B) \[x\log (\log x)-\frac{x}{\log x}+c\]
C) \[x\log (\log x)+\frac{\log x}{x}+c\]
D) \[x\log (\log x)-\frac{\log x}{x}+c\]
Correct Answer: B
Solution :
\[\int_{{}}^{{}}{\left[ \log (\log x)+\frac{1}{{{(\log x)}^{2}}} \right]\,dx}=\int_{{}}^{{}}{\log (\log x)dx+\int_{{}}^{{}}{\frac{1}{{{(\log x)}^{2}}}}}dx\] \[=x\log (\log x)-\int_{{}}^{{}}{\frac{x}{x\log x}\,dx+\int_{{}}^{{}}{\frac{1}{{{(\log x)}^{2}}}dx}}\] \[=x\log (\log x)-\frac{x}{\log x}-\int_{{}}^{{}}{\frac{1}{{{(\log x)}^{2}}}dx+\int_{{}}^{{}}{\frac{1}{{{(\log x)}^{2}}}dx}}\] \[=x\log (\log x)-\frac{x}{\log x}+c.\]
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