A) \[{{\log }_{e}}\left| \frac{{{x}^{3}}+1}{x} \right|+C\]
B) \[\frac{1}{2}{{\log }_{e}}\frac{{{({{x}^{3}}+1)}^{2}}}{|{{x}^{3}}|}+C\]
C) \[\frac{1}{2}{{\log }_{e}}\frac{|{{x}^{3}}+1|}{{{x}^{2}}}+C\]
D) \[{{\log }_{e}}\frac{|{{x}^{3}}+1|}{{{x}^{2}}}+C\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{2{{x}^{3}}-1}{{{x}^{4}}+x}}dx\] \[\int_{{}}^{{}}{\frac{2x-\frac{1}{{{x}^{2}}}}{{{x}^{2}}+\frac{1}{x}}}dx\] \[{{x}^{2}}+\frac{1}{x}=t\] \[\left( 2x-\frac{1}{{{x}^{2}}} \right)dx=dt\] \[\int_{{}}^{{}}{\frac{dt}{t}=\ell n(t)+C}\] \[=\ell n\left( {{x}^{2}}+\frac{1}{x} \right)+C\]
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