Why can't Mathematica solve this definite integral?

Questions
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1

Hi everyone,

I’m trying to compute the following definite integral (Mu is a parameter):

Integrate[Sqrt[3]/Sqrt[1 + Sqrt[1 + 12*u^2 - 24*\[Mu]]], {u, -Sqrt[1 + 8*\[Mu]]/2, Sqrt[1 + 8*\[Mu]]/2}]

Only for some specific case of Mu Mathematica seems to be able to compute it. The “funny” things are that:

  • if I keep the integral indefinite, it returns me a solution (quite ugly, but at least…):

Screenshot.jpg
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  • if I give precise values for the boundary (e.g, u=±1/2), after a long time it just returns the definite integral without any result.
  • if I additionally specify a precise value of Mu (so it knows Mu + boundary of integration), in one lucky case it is able to directly give me the result for the definite integral; this does not match with the value I would obtain by using the fundamental theorem of calculus (i.e. substituting the values of Mu and u in the ugly formula and taking the difference).

Has any of you an idea about what the problem could be? I would like to also point out that the square roots are always well definite for the values I consider.

Thank you.

2

The indefinite integral has a jump discontinuity at u=0, as you can see with a plot. You must be careful when applying the fundamental theorem of calculus. The reason why Mathematica has trouble with the definite integral may be that it cannot deal automatically with the branch cuts.