A) \[{{\tan }^{-1}}x+2{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\]
B) \[{{\tan }^{-1}}\left( \frac{x}{2} \right)-4{{\tan }^{-1}}x+C\]
C) \[-\frac{1}{3}{{\tan }^{-1}}x+\frac{2}{3}{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\]
D) \[4{{\tan }^{-1}}\left( \frac{x}{2} \right)-2{{\tan }^{-1}}x+C\]
Correct Answer: C
Solution :
[c] \[\int{\frac{{{x}^{2}}}{({{x}^{2}}+1)({{x}^{2}}+4)}}dx\]. \[=\frac{1}{3}\int{\left[ \frac{4}{{{x}^{2}}+4}-\frac{1}{{{x}^{2}}+1} \right]dx}\] \[=-\frac{1}{3}\int{\frac{1}{{{x}^{2}}+1}}dx+\frac{4}{3}\int{\frac{1}{{{x}^{2}}+4}dx}\] \[=-\frac{1}{3}{{\tan }^{-1}}x+\frac{4}{3}\times \frac{1}{2}{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\] \[=-\frac{1}{3}{{\tan }^{-1}}x+\frac{2}{3}{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\]
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