Hi.
I usually check my work with WolframAlpha and for this problem there seems to be a discrepency...
Here's what I've done:
Let
This means that
.
As for the limits with respect to u, we observe that
and .
Finally, evaluating:
Therefore,
But WAlpha gets something different.
The function you are integrating has a discontinuity over the interval of integration (at x = 2) so you have to take greater care in dealing with the resulting improper integrals.
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.
Of course that is true and I suggest to start with the indefinite integral...
(1)
Now we can devide the definite integral in two parts...
(2)
But both the integrals in (2) converge so that is...
(3)
So I agree with VonNemo and the result of WolframAlpha is difficult to undestand... at least for me...
Kind regards
I will elaborate on this statement.
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.