What is the integration of $(dx)/(x.sqrt(x^3+4))$ ??

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Answer 1 · 2018-03-07T00:35:15

$1/6 ln|{sqrt(x^3+4)-2}/{sqrt(x^3+4)+2}|+C$

Substitute $x^3+4=u^2$ . Then $3x^2dx=2udu$ , so that

$dx/{x sqrt{x^3+4}} = {2udu}/{3x^3u} = 2/3 {du}/(u^2-4) = 1/6({du}/{u-2}-{du}/{u+2})$

Thus

$int dx/{x sqrt{x^3+4}} = 1/6 int ({du}/{u-2}-{du}/{u+2})=1/6 ln|{u-2}/{u+2}|+C$
$=1/6 ln|{sqrt(x^3+4)-2}/{sqrt(x^3+4)+2}|+C$

What is the integration of (dx)/(x.sqrt(x^3+4))(dx)/(x.sqrt(x^3+4))??