Question

If $\underset{1}{\overset{x}{\int }}\frac{dt}{|t\sqrt{t^2-1}}=\frac{\pi }{6}$ , then $x$ can be equal to

• A. $\frac{2}{\sqrt{3}}$
• B. $\sqrt{3}$
• C. $2$
• D. None of these

Answer 1

Answer: (A)

$\underset{1}{\overset{x}{\int }}\frac{dt}{|t|\sqrt{t^2-1}}=\frac{\pi }{6}$
$\Rightarrow {\left[{\sec }^{-1}t\right]}_1^x=\frac{\pi }{6}$
$\Rightarrow {\sec }^{-1}x-{\sec }^{-1}1=\frac{\pi }{6}$
$\Rightarrow {\sec }^{-1}x-0=\frac{\pi }{6}$
$\Rightarrow x=\sec \frac{\pi }{6}$
$\Rightarrow x=\frac{2}{\sqrt{3}}$