A) \[2\log (2+\sqrt{5})\]
B) \[\log (2+\sqrt{5})\]
C) \[\frac{1}{\sqrt{5}}\log (2+\sqrt{5})\]
D) \[\frac{1}{2}\log (2+\sqrt{5})\]
Correct Answer: B
Solution :
\[\left| \frac{\log \left( t+\sqrt{1+{{t}^{2}}} \right)}{\sqrt{1+{{t}^{2}}}} \right.dt=\frac{1}{2}{{\left( g(t) \right)}^{2}}+C\] Differentiating both sides \[\frac{\log \left( t+\sqrt{1+{{t}^{2}}} \right)}{\sqrt{1+{{t}^{2}}}}=g(t)g'(t)\] \[\Rightarrow \]\[g(t)=\log \left( t+\sqrt{1+{{t}^{2}}} \right)\] \[\therefore \]\[g(2)=\log \left( 2+\sqrt{5} \right)\]
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