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I have a following case to integrate: $$\int{\sqrt{1+\left(\frac12x-\frac1{2x}\right)^2}dx}$$

I tried following the steps that are suggested for integrating square roots with enclosed sum of quadratics, but I am having trouble with the substitution, due to the $\frac1{2x}$ part.

I tried calculating the square before doing the substitution, but the fraction that is causing the problems with substitution remains.

This is what I used to look for integration methods: http://tutorial.math.lamar.edu/Classes/CalcII/IntegrationStrategy.aspx

I tried following the suggestions from this video: https://www.youtube.com/watch?v=23DbI7ZHOwY but I do not have simple $x^2$, and I can't find simple substitution that would transform it into such.

Any pointers would be much appreciated.

Edit: After a nice hint from Tired I noticed, that this can be written as complete square $\frac14(x+x^{-1})$ and then the solution becomes trivial and no susbstitution is required at all. Thanks!

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    $\begingroup$ Did you consider trig substitution? $\endgroup$ – Varun Iyer Jan 10 '16 at 13:45
  • $\begingroup$ i would suggest a substitution $x=e^y$ together with the fact that $\cosh^2(y)-\sinh^2(y)=1$ $\endgroup$ – tired Jan 10 '16 at 13:48
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hint: for the integrand we get $$\frac{x^4+2x^2+1}{4x^2}$$

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  • $\begingroup$ Nice observation (+1) $\endgroup$ – tired Jan 10 '16 at 14:06

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