What is the integral of x(modx)? Here mod is modulus of x.

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Answer 1 · 2018-04-21T20:42:55

See below

Let #x = - alpha# where #alpha > 0#

#int \ x |x| \ dx#

#= int \ - alpha |- alpha| \ d(- alpha)#

#= int \ alpha^2 \ d alpha #

#= alpha^3 /3 + C#

#= - x^3/3 + C#

Now let #x = beta# where #beta > 0#

#int \ x |x| \ dx#

#= int \ beta |beta| \ d(beta)#

#= int \ beta^2 \ d beta #

#= beta^3 /3 + C#

#= x^3/3 + C#

If you know the signum function you can write this as:

#int \ x |x| \ dx = sgn(x) x^3/3 + C#