A) \[\frac{1}{4}\log \frac{32}{17}\]
B) \[\frac{1}{2}\log \frac{32}{17}\]
C) \[\frac{1}{4}\log \frac{16}{17}\]
D) None of these
Correct Answer: A
Solution :
Here \[\varphi (x)=\frac{1}{x({{x}^{4}}+1)}=\frac{1}{x}-\frac{{{x}^{3}}}{{{x}^{4}}+1}\] Þ \[\int_{1}^{2}{\varphi (x)dx=\int_{1}^{2}{\,\left( \frac{1}{x}-\frac{{{x}^{3}}}{{{x}^{4}}+1} \right)}\,dx}\] \[=|\log x|_{1}^{2}-\left| \frac{1}{4}\log ({{x}^{2}}+1) \right|_{1}^{2}=\frac{1}{4}\log \frac{32}{17}\].
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