OK, maybe it’s not what you think, the CAS‘s of the world are not going to rise up and enslave us :-P (Not yet, anyway.) But it is always a good idea to view their results with a healthy dose of skepticism, because like any computer program, they have bugs, and every once in a while one can actually mess up your calculation.

Here’s my example:

Plug this into Mathematica 8.0 and it will tell you that the integral is equal to \(2\pi i J_1(k)\), where \(J_1\) is a Bessel function of the first kind:

In[1]  := Integrate[Cos[x] Exp[I k Cos[x - y]], {x, 0, 2 Pi}]
Out[1] := ConditionalExpression[2 I \[Pi] BesselJ[1, k], k \[Element] Reals]

But it’s not! If you shift the integration variable \(x\to x + y\), Mathematica tips its hand and tells you the real answer, \(2\pi i J_1(k)\cos y\):

In[2]  := Integrate[Cos[x + y] Exp[I k Cos[x]], {x, 0, 2 Pi}]
Out[2] := 2 Pi I BesselJ[1, k] Cos[y]

You can also do …