A) \[2(\sqrt{x}+{{\tan }^{-1}}\sqrt{x})+c\]
B) \[2(\sqrt{x}+{{\cot }^{-1}}\sqrt{x})+c\]
C) \[2(\sqrt{x}-{{\cot }^{-1}}-\sqrt{x})+c\]
D) \[2(\sqrt{x}-ta{{n}^{-1}}\sqrt{x})+c\]
Correct Answer: D
Solution :
Let \[I=\int{\frac{\sqrt{x}\sqrt{x}}{\sqrt{x}\,(x+1)}}dx\] \[=\int{\frac{x+1}{\sqrt{x}(x+1)}}dx-\int{\frac{1}{\sqrt{x}(x+1)}}dx\] \[=\int{\frac{1}{\sqrt{x}}}dx-\int{\frac{1}{\sqrt{x}(x+1)}}\,dx\] \[=2{{x}^{1/2}}-2{{\tan }^{-1}}\sqrt{x}+c\] \[=2(\sqrt{x}-{{\tan }^{-1}}\sqrt{x})+c\]
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