TITLE: How can we find $\int\frac{1}{x^2\sqrt{x^2+1}}dx$?
QUESTION [5 upvotes]: How can we integrate 
$$\int\frac{1}{x^2\sqrt{x^2+1}}dx$$
I tried some substitutions but I failed . I tried for example $$v=\sqrt{1+x^2}$$

REPLY [8 votes]: Another way to evaluate the integral, without making a trigonometric or hyperbolic substitution, is to make the substitution $x=1/t$.  

Proceeding reveals 
$$\begin{align}
\int \frac{1}{x^2\sqrt{1+x^2}}\,dx&\overbrace{=}^{x=1/t}-\int \frac{t}{\sqrt{t^2+1}}\,dt\\\\
&=-\sqrt{1+t^2}+C\\\\
&\overbrace{=}^{t=1/x}-\frac{\sqrt{1+x^2}}{x}+C
\end{align}$$
And we are done!