A) \[x{{\sec }^{-1}}x+{{\cosh }^{-1}}x+C\]
B) \[x{{\sec }^{-1}}x-{{\cosh }^{-1}}x+C\]
C) \[x{{\sec }^{-1}}x-{{\sin }^{-1}}x+C\]
D) None of these
Correct Answer: B
Solution :
\[I=\int{{{\cos }^{-1}}\left( \frac{1}{x} \right)\,dx}\]\[=\int{{{\sec }^{-1}}x.1\,\,dx}\] \[={{\sec }^{-1}}x\int{dx-\int{\left[ \frac{d}{dx}{{\sec }^{-1}}x\int{dx\,} \right]}\,dx}\] \[=x{{\sec }^{-1}}x-\int{\frac{1}{x\sqrt{{{x}^{2}}-1}}x.\,\,dx}\] \[=x{{\sec }^{-1}}x-\int{\frac{1}{\sqrt{{{x}^{2}}-1}}dx}\]\[=x{{\sec }^{-1}}x-{{\cosh }^{-1}}x+c\].
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