A) \[-\sqrt{2}{{\tan }^{-1}}x+\sqrt{3}{{\tan }^{-1}}x+c\]
B) \[-\sqrt{2}{{\tan }^{-1}}\frac{x}{\sqrt{2}}+\sqrt{3}{{\tan }^{-1}}\frac{x}{\sqrt{3}}+c\]
C) \[\sqrt{2}{{\tan }^{-1}}\frac{x}{\sqrt{2}}+\sqrt{3}{{\tan }^{-1}}\frac{x}{\sqrt{3}}+c\]
D) None of these
Correct Answer: B
Solution :
\[\int_{{}}^{{}}{\frac{{{x}^{2}}}{({{x}^{2}}+2)({{x}^{2}}+3)}}\,dx=\int_{{}}^{{}}{\left[ \frac{3}{{{x}^{2}}+3}-\frac{2}{{{x}^{2}}+2} \right]}\,dx\] \[=\frac{3}{\sqrt{3}}{{\tan }^{-1}}\frac{x}{\sqrt{3}}-\frac{2}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{x}{\sqrt{2}} \right)+c\] \[=\sqrt{3}{{\tan }^{-1}}\left( \frac{x}{\sqrt{3}} \right)-\sqrt{2}{{\tan }^{-1}}\left( \frac{x}{\sqrt{2}} \right)+c.\]
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