TITLE: How to integrate $\int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}}$?
QUESTION [6 upvotes]: How to integrate $$\int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}}$$
I tried both $t=\sqrt{x^2-1}$ and $t=\sin x$ but didn't reach the right result.

REPLY [3 votes]: $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
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\begin{align}&\overbrace{\color{#66f}{\large%
\int_{1}^{\infty}{\dd x \over x^{2}\root{x^{2} - 1}}}}
^{\ds{\dsc{x} = \dsc{1 \over t}\ \imp\ \dsc{t} = \dsc{1 \over x}}}\ =\
\int_{1}^{0}{-\,\dd t/t^{2} \over t^{-2}\root{1/t^{2} - 1}}
=\int_{0}^{1}{t\,\dd t \over \root{1 - t^{2}}}
=\left. -\root{1 - t^{2}}\right\vert_{0}^{1}=\color{#66f}{\LARGE 1}
\end{align}