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