Check this out. I hand-evaluated this integral and my pretty sure the answer is zero, but Wolfram returns the value $4i\pi$ instead.
2026-03-28 03:32:55.1774668775
Getting weird integral evaluation from Wolfram Alpha
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I agree with WA. The first piece is zero. For the second, let $z=e^{i t}$ and work the following contour integral:
$$\begin{align} 6 \oint_{|z|=1} \frac{dz}{2+3 z} = 2 \oint_{|z|=1} \frac{dz}{(2/3)+z}\end{align}$$
The integral is equal to $i 2 \pi$ times the residue at the pole $z=-2/3$, which is inside the unit circle. The result follows.