MHB Integrating the Square Root of a Fraction with a Radical

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The discussion focuses on evaluating the integral function \( I_5 = \int \sqrt{\frac{x^2-4}{x}} \, dx \). Participants noted that a simple substitution method does not yield a convenient solution, and Wolfram Alpha (W|A) indicates a non-elementary result. Additionally, it was highlighted that the function is undefined at \( x=0 \), prompting a suggestion to consider the alternative integral \( \int \frac{\sqrt{x^2-4}}{x}\,\mathrm{d}x \) for potentially easier evaluation.

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$\tiny{10.08.06}\\$
$\textsf{Evaluate the function}$
\begin{align*}\displaystyle
I_5&=\int \sqrt{\frac{x^2-4}{x}} \, dx
\end{align*}

ok, I thought this would be a simple U subst, but nothing looks convienent
 
Last edited:
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W|A gives a non-elementary result.
 
I presume when $x=0$ it is undefined:cool:
 
Are you sure you did not mean $\int \frac{\sqrt{x^2-4}}{x}\,\mathrm{d}x$?
 

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