A) \[\frac{\log x}{{{(\log x)}^{2}}+1}+c\]
B) \[\frac{x}{{{x}^{2}}+1}+c\]
C) \[\frac{x{{e}^{x}}}{1+{{x}^{2}}}+c\]
D) \[\frac{x}{{{(\log x)}^{2}}+1}+c\]
Correct Answer: D
Solution :
Now \[\frac{d}{dx}\left( \frac{x}{{{(\log x)}^{2}}+1}+c \right)\] \[=\frac{{{(\log x)}^{2}}+1-x\left( 2\log x.\frac{1}{x} \right)}{{{[{{(\log x)}^{2}}+1]}^{2}}}\] \[=\frac{{{(\log x)}^{2}}+1-2\log x}{{{[{{(\log x)}^{2}}+1]}^{2}}}\] \[=\frac{{{(\log x-1)}^{2}}}{{{[{{(\log x)}^{2}}+1]}^{2}}}={{\left( \frac{\log x-1}{{{(\log x)}^{2}}+1} \right)}^{2}}\] Hence, option is correct.
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