Question

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

Answer 1

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

Explanation:

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`