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