Answer
$arcsec (x)+\dfrac{2\pi}{3}$
Work Step by Step
We need to integrate:
$y=\int \dfrac{dx}{x\sqrt{x^2-1}}$
We use the differential trigonometric formula:
$\int \dfrac{dx}{x\sqrt{x^2-a^2}}=arcsec (x/a)+C$
$y=arcsec (x)+c$
Take initial conditions: $y(2)=\pi$
Then $y(2)=arcsec (2)+c =\pi \implies c=\dfrac{2\pi}{3}$
Thus, we have $y=arcsec (x)+\dfrac{2\pi}{3}$
