Definite integration yields a strange answer. (Solved)

[parisse]

You misunderstand the meaning of the complex mode setting (note that this is named i on the Prime CAS settings, complex on the Prime is the complex variable mode setting). If you keep complex mode unchecked, it does not mean you will stay in the real domain. For example, sqrt(-1) will return i if complex is unchecked. Checking complex just says to the CAS that factor and associated functions should behave like if coefficients were complex even if they are all reals.
On the hp49, it was mandatory to check complex if you wanted to leave the real domain. It was however not efficient because you had to switch mode too often. That's the main reason I designed giac that way.
For fractional powers, you must understand that x^(1/3) *is* a complex number if x<0, whatever the complex mode setting is (on the hp49, the system would ask you to switch to complex mode or error, on the Prime it will return a complex number). If you want the real 3rd root, just use NTHROOT or surd.