Trignometric Integration Question

I'm stuck with this integral

$\displaystyle \int_1^2 \sqrt{(x^2-1)} dx$

I substitute.
$[br]x = sec(u)[br]dx = sec(u)tan(u) du[br]$

Getting:
$[br]\displaystyle \int_1^2\ tan^2(u)sec(u) du[br]$

From here I get stuck. I've tried various substitutions but I can't seem to get close to making it work.

Any ideas?

Thank for you for any assistance.

I would have used x = coshu
It is much easier

So it is. Thanks, that did make it much simpler.

You can use a simple hyperbolic substitution here, have you covered hyperbolic functions or is this a C4 question?

$\cosh^2 \theta -1 = \sinh^2 \theta$

$x = \cosh \theta \to dx = \sinh\theta \hspace{1 mm} d\theta$

Do it yourself from here.

As others have said, there are substitutions you can make that would make this much easier.
Don't forget to take care with the limits of your integration, though! Are you sure they'd be still be 1 and 2 after that substitution?

This isn't a C4 question. I'm an undergraduate minoring in maths, I haven't covered hyperbolic functions as much as I'd like. Since I didn't take further maths, so I need to do some leg work here evidently with considering them in relation to integration.

Thanks for the help, as stated this substitution did make things much simpler.