A) \[\ln \,\left| x-\sqrt{{{x}^{2}}-1} \right|-{{\tan }^{-1}}x+c\]
B) \[\ln \,\left| x+\sqrt{{{x}^{2}}-1} \right|-{{\tan }^{-1}}x+c\]
C) \[\ln \,\left| x-\sqrt{{{x}^{2}}-1} \right|-{{\sec }^{-1}}x+c\]
D) \[\ln \,\left| x+\sqrt{{{x}^{2}}-1} \right|-{{\sec }^{-1}}x+c\]
Correct Answer: D
Solution :
[d] \[I=\int{\frac{\sqrt{x-1}}{x\sqrt{x+1}}dx}\] \[=\int{\frac{x-1}{x\sqrt{{{x}^{2}}-1}}dx}\] \[=\int{\frac{dx}{\sqrt{{{x}^{2}}-1}}-\int{\frac{dx}{x\sqrt{{{x}^{2}}-1}}}}\] \[=\ln |x+\sqrt{{{x}^{2}}-1}|-{{\sec }^{-1}}x+c\]
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