A) \[\frac{1}{4}\log \frac{17}{32}\]
B) \[\frac{1}{4}\log \frac{17}{2}\]
C) \[\log \frac{17}{2}\]
D) \[\frac{1}{4}\log \frac{32}{17}\]
Correct Answer: D
Solution :
\[\int_{1}^{2}{\frac{dx}{x(1+{{x}^{4}})}=\int_{1}^{2}{\frac{dx}{{{x}^{5}}\left( 1+\frac{1}{{{x}^{4}}} \right)}}}\] Put \[\left( 1+\frac{1}{{{x}^{4}}} \right)=z\Rightarrow \frac{-4}{{{x}^{5}}}dx=dz\] Þ \[\frac{-1}{4}\int_{2}^{17/16}{\frac{dz}{z}=\left[ \frac{-1}{4}\log z \right]_{2}^{17/16}}\]=\[\frac{1}{4}\log 2-\frac{1}{4}\log \frac{17}{16}\] Þ \[I=\frac{1}{4}\log \left( \frac{32}{17} \right)\].
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