A) \[x{{\sec }^{-1}}x+{{\cosh }^{-1}}x+c\]
B) \[x{{\sec }^{-1}}x-{{\cosh }^{-1}}x+c\]
C) \[{{\cosh }^{-1}}x-{{\sec }^{-1}}x+c\]
D) None of the above
Correct Answer: B
Solution :
\[\int{{{\cos }^{-1}}\left( \frac{1}{x} \right)}\,dx\] \[\int{\underset{I}{\mathop{{{\cos }^{-1}}}}\,\left( \frac{1}{x} \right)}\,.\underset{II}{\mathop{1}}\,dx\] \[={{\cos }^{-1}}\left( \frac{1}{x} \right).x-\int{\left( \frac{-1}{\sqrt{1-{{\left( \frac{1}{x} \right)}^{2}}}} \right)}.\left( -\frac{1}{{{x}^{2}}} \right)x\,dx\] \[=x{{\cos }^{-1}}\left( \frac{1}{x} \right)-\int{\frac{1}{\frac{x\sqrt{{{x}^{2}}-1}}{x}}}dx\] \[=x{{\sec }^{-1}}x-\int{\frac{1}{\sqrt{{{x}^{2}}-1}}}dx\] \[=x{{\sec }^{-1}}x-{{\cosh }^{-1}}x+c\] \[\left[ \because \int{\frac{1}{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx={{\cosh }^{-1}}\left( \frac{x}{a} \right)+c} \right]\]
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