Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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
add comment

2 Answers 2

up vote 5 down vote accepted

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

share|improve this answer
    
Now I feel stupid, thanks! –  McT Nov 20 '12 at 1:16
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.