A) \[-\sqrt{2}{{\tan }^{-1}}\left( \frac{x}{\sqrt{2}} \right)+\sqrt{3}{{\tan }^{-1}}\left( \frac{x}{\sqrt{3}} \right)+C\]
B) \[\sqrt{2}{{\tan }^{-1}}\left( \frac{x}{\sqrt{2}} \right)+\sqrt{3}{{\tan }^{-1}}\left( \frac{x}{\sqrt{3}} \right)+C\]
C) \[-\sqrt{2}{{\tan }^{-1}}x+\sqrt{3}{{\tan }^{-1}}x+C\]
D) None of the above
Correct Answer: A
Solution :
\[\int{\frac{{{x}^{2}}}{({{x}^{2}}+2)({{x}^{2}}+3)}dx=\int{\left( \frac{3}{{{x}^{2}}+3}-\frac{2}{{{x}^{2}}+2} \right)dx}}\] \[=\frac{3}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{x}{\sqrt{3}} \right)-\frac{2}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{x}{\sqrt{2}} \right)+C\] \[=\sqrt{3}{{\tan }^{-1}}\left( \frac{x}{\sqrt{3}} \right)-\sqrt{2}{{\tan }^{-1}}\left( \frac{x}{\sqrt{2}} \right)+C\]
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