Determine whether $\int^{\infty}_{2} \frac{1}{x\sqrt{x^2-4}} dx$ converges. Determine whether $$\int^{\infty}_{2} \frac{1}{x\sqrt{x^2-4}} dx$$ converges.
Doing some rough work, I realize that this function near $\infty$, behaves like $$\frac{1}{x\sqrt{x^2}} = \frac{1}{x^2}$$
I know this function converges, but I am having a hard time finding a $larger$ function that converges too.
I know that:
$$\frac{1}{x\sqrt{x^2-4}} \leq \frac{1}{\sqrt{x^2-4}}$$
but the bigger function diverges, so it doesn't work.

Easier and more accurate to evaluate directly. I took a look at that integral and it reminded me of something to do with secant or was that inverse secant? Tried an integral table quick search but failed to find the one needed. Paper and pen a couple of minutes gives:
By trig substitution
Let $x/2 = \sec u;\ \  x = 2 \sec u$
$dx = 2 \sec u \tan u \ du; \ \  \sqrt{x^2 - 4} = \sqrt{4\sec^2 u -4} = 2 \tan u $ 
$$ \int\frac{dx}{x\sqrt{x^2 - 4}} = \int \frac{2 \sec u \tan u \ du}{ 2 \sec u \ 2 \tan u} = \frac{1}{2}\int du = u = \frac{1}{2}\sec^{-1}(\frac{x}{2}) $$
$$ \int_2^{\infty}\frac{dx}{x\sqrt{x^2 - 4}}=\lim_{b \rightarrow \infty} \frac{1}{2}\sec^{-1}(\frac{x}{2})]_{x=2}^b   $$
$$ = \frac{1}{2}(\lim_{b \rightarrow \infty} \sec^{-1}(\frac{b}{2}) - \sec^{-1}(\frac{2}{2}))   $$
$$ = \frac{1}{2} (\pi/2 - 0) = \pi/4$$
So yes it converges and we have the limit.
