I usually check my work with WolframAlpha and for this problem there seems to be a discrepency...
Here's what I've done:
This means that
As for the limits with respect to u, we observe that
But WAlpha gets something different.
Further, WolframAlpha does not give a real value to (-1)^(2/3) (nor does a CAS calculator. There are technical reasons for this). There are three complex values, namely , and . Since your integral is real (why?) ......
Edit: Fixed a careless typo. Misread cis(-pi/2) = -1 instead of cis(-pi) = -1.
Now we can devide the definite integral in two parts...
But both the integrals in (2) converge so that is...
So I agree with VonNemo and the result of WolframAlpha is difficult to undestand... at least for me...
When the TI-89 CAS calculator is operating in real mode it uses the real branch (when it exists) for fractional powers that have a reduced exponent with odd denominator. So for (-1)^(2/3) it returns a value of 1.
However, when operating in complex mode, or when the real branch does not exist, the TI-89 uses the principle branch. Thus, for (-1)^(2/3) it returns a value of (which is the rectangular form of , by the way). No doubt something similar is happening with Mathematica: (-1)^(2/3) - Wolfram|Alpha
and this is what's causing the trouble with regards to the output for the definite integral that initiated this thread.