A) \[\log (x+\sqrt{x-1})+{{\sin }^{-1}}\left( \sqrt{\frac{x-1}{x}} \right)+c\]
B) \[\log (x+\sqrt{x-1})+c\]
C) \[\log (x+\sqrt{x-1})-\frac{2}{\sqrt{3}}{{\tan }^{-1}}\]
D) \[\left( \frac{2\sqrt{x-1}+1}{\sqrt{3}} \right)\] none of the above
Correct Answer: C
Solution :
Let \[I=\int_{{}}^{{}}{\frac{dx}{x+\sqrt{x-1}}}\] Let \[x={{t}^{2}}+1\Rightarrow dx=2t\,dt\] \[\therefore \] \[I=\int_{{}}^{{}}{\frac{2t}{{{t}^{2}}+t+1}}dt\] \[=\int_{{}}^{{}}{\frac{2t+1}{{{t}^{2}}+t+1}}dt-\int_{{}}^{{}}{\frac{1}{{{t}^{2}}+t+1}}dt\] \[=\log ({{t}^{2}}+t+1)-\int_{{}}^{{}}{\frac{1}{{{\left( t+\frac{1}{2} \right)}^{2}}+{{\left( \frac{\sqrt{3}}{2} \right)}^{2}}}dt}\] \[=\log (x+\sqrt{x-1})-\frac{2}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\sqrt{x-1}+1}{\sqrt{3}} \right)+c\]
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