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

1 Answer
Mar 7, 2018

#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#