the important thing is to be able to recognize when you can use it. the method is essentially the same in all cases.
for my method:
$\displaystyle x^2 = \sqrt{3} \sec t$
$\displaystyle \Rightarrow 2x~dx = \sqrt{3} \sec t \tan t ~dt$
$\displaystyle \Rightarrow x~dx = \frac {\sqrt{3}}2 \sec t \tan t ~dt$
so our integral becomes:
$\displaystyle \frac {\sqrt{3}}2 \int \frac {\sec t \tan t}{\sqrt{3 \sec^2 t - 3}}~dt$
this simplifies to
$\displaystyle \frac 12 \int \sec t~dt$
now can you finish up? don't forget this is a definite integral, so you have to deal with the limits at some point