A) \[\log \left[ \frac{{{x}^{2}}+1}{{{x}^{2}}+3} \right]+c\]
B) \[\log \left[ \frac{{{x}^{2}}+3}{{{x}^{2}}+1} \right]+c\]
C) \[{{\tan }^{-1}}x+\left( \frac{1}{\sqrt{3}} \right){{\tan }^{-1}}\left( \frac{x}{\sqrt{3}} \right)+c\]
D) \[\left( \frac{4}{\sqrt{3}} \right){{\tan }^{-1}}x{{\tan }^{-1}}\left( \frac{x}{\sqrt{3}} \right)+c\]
E) \[2\log ({{x}^{2}}+1)({{x}^{2}}+3)+c\]
Correct Answer: A
Solution :
Let \[I=\int{\frac{4x}{({{x}^{2}}+1)({{x}^{2}}+3)}}dx=\int{\frac{2x}{{{x}^{2}}+1}}dx\] \[-\int{\frac{2x}{{{x}^{2}}+3}}dx\] Let\[{{x}^{2}}+1=u\]and \[{{x}^{2}}+3=v\] \[\Rightarrow \] \[2x\text{ }dx=du\] and \[2x\text{ }dx=dv\] \[\therefore \] \[I=\int{\frac{du}{u}}-\int{\frac{du}{v}}\] \[=log\text{ }u-log\text{ }v+c\] \[=\log \frac{({{x}^{2}}+1)}{({{x}^{2}}+3)}+c\]
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