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