A) \[x-4{{\tan }^{-1}}\left( \frac{x}{2} \right)+c\]
B) \[x-2{{\tan }^{-1}}\left( \frac{x}{2} \right)+c\]
C) \[x+4{{\tan }^{-1}}\left( \frac{x}{2} \right)+c\]
D) None of the above
Correct Answer: B
Solution :
Let \[I=\int{\frac{{{x}^{2}}dx}{{{x}^{2}}+4}}\] \[\Rightarrow \] \[I=\int{\frac{{{x}^{2}}+4-4}{{{x}^{2}}+4}dx}\] \[\Rightarrow \] \[I=\int{\frac{{{x}^{2}}+4}{{{x}^{2}}+4}}dx-\int{\frac{4}{{{x}^{2}}+4}dx}\] \[\Rightarrow \] \[I=\int{1\,}dx-4\int{\frac{1}{{{x}^{2}}+4}dx}\] \[\Rightarrow \] \[I=x-4.\frac{1}{2}{{\tan }^{-1}}\left( \frac{x}{2} \right)+c\] \[\Rightarrow \] \[I=x-2{{\tan }^{-1}}\left( \frac{x}{2} \right)+c\]
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