I am trying to find $\int_\sqrt{2}^2 \frac{dt}{t^2 \sqrt{t^2-1}}$
$t = \sec \theta$ $dt = \sec \theta \tan\theta $
$\int_\sqrt{2}^2 \frac{dt}{\sec ^2 \theta \sqrt{\sec^2 \theta-1}}$
$\int_\sqrt{2}^2 \frac{dt}{\sec ^2 \theta \tan^2 \theta}$
$\int_\sqrt{2}^2 \frac{\sec \theta \tan\theta}{\sec ^2 \theta \tan^2 \theta}$
$\int_\sqrt{2}^2 \frac{1}{\sec \theta}$
$\int_\sqrt{2}^2 \cos \theta$
$\sin \theta$
Then I need to make it in terms of t.
$t = \sec \theta$
So I just use the arcsec which is
$\theta =\operatorname{arcsec} t$
$\sin (\operatorname{arcsec} t)$
This is wrong but I am not sure why.