curve length with ln function

Find the length of the curve:

$\displaystyle y = \frac{x^2}{4} - \ln \sqrt x $ for $\displaystyle 1\leq x\leq 2

$

If I understand correctly, Length = integral from 1 to 2 of $\displaystyle \sqrt {1+(\frac{dy}{dx})^2} dx $

So in this case, I need to find y' and substitute it in the formula and then find the integral.

y =$\displaystyle \frac{x}{2} - \frac{1}{2}\frac{1}{x} ==> \frac{x^2-1}{2x} $

Then I plug in y' into the formula to get (integral from 1 to 2)

L =$\displaystyle \int \sqrt{1 + (\frac{x^2-1}{2x})^2} dx

$ I'm not sure where to go from here to make the integration work. I've been stuck on this for a while, so if anyone could help, I'd appreciate it!