A) \[{{\tan }^{-1}}\sqrt{4x+3}+c\]
B) \[3{{\tan }^{-1}}\sqrt{4x+3}+c\]
C) \[2{{\tan }^{-1}}\sqrt{4x+3}+c\]
D) \[4{{\tan }^{-1}}\sqrt{4x+3}+c\]
Correct Answer: C
Solution :
Let\[I=\int{\frac{dx}{(x+1)\sqrt{4x+3}}}\] Put\[4x+3={{t}^{2}}\Rightarrow 4dx=2t\,\,dt\] \[\therefore \]\[I=\frac{1}{2}\int{\frac{t\,\,dt}{\left( \frac{{{t}^{2}}-3}{4}+1 \right)}=2\int{\frac{dt}{1+{{t}^{2}}}}}\] \[2{{\tan }^{-1}}t+c=2{{\tan }^{-1}}\sqrt{4x+3}+c\]
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