MHB Stumped by Definite Integral: $\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx$

[Bushy]

Was asked to solve this definite integral in a tech free test. Not sure how to go about it.

$$\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx.$$

I know here is a relationship between inverse sin and the sqrt function but with just sin x?

 

[Ackbach]

Well, Mathematica balks at evaluating it, so it seems to me it is a rather non-trivial integral. I tried the substitution $y=\arccsc(x)$, which changes the integral to
$$\int_{\pi/6}^{\pi/2}\sin(\csc(y)) \, dy.$$
Problem is, this new integral is no easier than the old one. The numerical value is about $0.9597$. If you're not allowed any technology, I don't see how you could arrive at any solution.