A) \[\log \sqrt{1+x}-\frac{1}{2}\log \sqrt{1+{{x}^{2}}}+\frac{1}{2}{{\tan }^{-1}}x+c\]
B) \[\log \sqrt{1+x}-\log \sqrt{1+{{x}^{2}}}+{{\tan }^{-1}}x+c\]
C) \[\log \sqrt{1+{{x}^{2}}}-\log \sqrt{1+x}+\frac{1}{2}{{\tan }^{-1}}x+c\]
D) \[\log \sqrt{1+x}+{{\tan }^{-1}}x+\log \sqrt{1+{{x}^{2}}}+c\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{dx}{1+x+{{x}^{2}}+{{x}^{3}}}=\int_{{}}^{{}}{\frac{dx}{(1+x)(1+{{x}^{2}})}}}\] \[=\frac{1}{2}\int_{{}}^{{}}{\frac{1}{1+{{x}^{2}}}\,dx}+\frac{1}{2}\int_{{}}^{{}}{\frac{1}{1+x}\,dx}-\frac{1}{2}\int_{{}}^{{}}{\frac{x}{1+{{x}^{2}}}\,dx}\] \[=\frac{1}{2}{{\tan }^{-1}}x+\log \sqrt{1+x}-\frac{1}{2}\log \sqrt{1+{{x}^{2}}}+c\].
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