A) \[{{\tan }^{-1}}\sqrt{x}+C\]
B) \[2{{\tan }^{-1}}x+C\]
C) \[2{{\tan }^{-1}}(\sqrt{x})+C\]
D) \[{{\tan }^{-1}}\left( {{x}^{\frac{3}{2}}} \right)+C\]
E) \[2{{\tan }^{-1}}\left( {{x}^{\frac{3}{2}}} \right)+C\]
Correct Answer: C
Solution :
\[\int{\frac{dx}{(x+1)\sqrt{x}}}\] Put \[x={{t}^{2}},2t\,dt=dx\] \[\Rightarrow \] \[\int{\frac{2t\,dt}{({{t}^{2}}+1)t}}\] \[\int{\frac{2\,dt}{(1+{{t}^{2}})}}\] \[=2{{\tan }^{-1}}t+C\] \[=2{{\tan }^{-1}}\sqrt{x}+C\]
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