# Integral $\int \sqrt{4+4x^2+1/x^2} dx$

I'm totally lost on how I would integrate the following function:

$f(x) = \sqrt{4+4x^2+1/x^2}$

If anyone could even just point me to the method of integration that would be grand.

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If I have understood it correctly (see my edit), you could notice that it is a perfect square. –  Ross Millikan Nov 20 '12 at 1:14
You are probably solving an arclength problem. Typically, when you have a function $y$, differentiate, square the derivative, add $1$, take the square root, you get an awful mess that can't be integrated in elementary terms. But with very careful choice of constants, one can produce artificial situations where things magically simplify. –  André Nicolas Nov 20 '12 at 1:25
Note that it is non-trivial to compute general integrals of this form when the function under the radical is not a quadratic perfect square of linear factors. –  tacos_tacos_tacos Nov 20 '12 at 1:26

$$4 + 4x^2 + \dfrac1{x^2} = \left(2x + \dfrac1x \right)^2$$ Hence, \begin{align} \sqrt{4 + 4x^2 + \dfrac1{x^2}} & = \begin{cases} 2x + \dfrac1x & x > 0\\ -\left( 2x + \dfrac1x\right) & x < 0\end{cases}\\ & = 2 \vert x \vert + \dfrac1{\vert x \vert} \end{align}
Compute the square root!${}{}{}{}{}$