Question

$\int \frac{1}{x}\sqrt{\frac{x-1}{x+1}}dx=$

• A. ${\cosh }^{-1}x-{\sec }^{-1}x+c$
• B. ${\cosh }^{-1}x+{\sec }^{-1}x+c$
• C. ${\sinh }^{-1}x-{\sec }^{-1}x+c$
• D. ${\sinh }^{-1}x-{\csc }^{-1}x+c$

Answer 1

The correct option is A ${\cosh }^{-1}x-{\sec }^{-1}x+c$

Let $I=\int \frac{1}{x}\sqrt{\frac{x-1}{x+1}}dx$

$=\int \frac{x-1}{x\sqrt{x^2-1}}dx$ .......... ratinalisation

$=\int \frac{1}{\sqrt{x^2-1}}dx-\int \frac{1}{x\sqrt{x^2-1}}dx$

$I=cos{h}^{-1}x-{\sec }^{-1}x+C$