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

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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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2 Answers

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$$4 + 4x^2 + \dfrac1{x^2} = \left(2x + \dfrac1x \right)^2$$ 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}

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Compute the square root!${}{}{}{}{}$

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