Hello, thanks for the help a few days ago, I found a way to solve this problem and I came back to give some closure. If we use sin(x)=(e^ixe^ix)/2i, and multiply that by itself four times (to the power 4), we will get some positive and some negative exponentials. There is a pole at z=0. We need to consider 2 contours, one for the positive exponentials and one for the negative exponentials.
For the positive exponentials, the contour is above the real axis and goes around the pole at z=0. Since there is no residue in here, the integral is 0.
For the negative exponentials, the contour is below the real axis and has the a residue inside. Therefore, to compute the integral of sinc(x)^4 between negative infinity and infinity, we just need to find the value of the residue in the lower contour. Using the formula for a residue of a pole of order 4:
Residue=2ipi * lim(z>0) of (1/((41)!))*(dł/dzł)(((z0)^4)*(negative exponentials))
We find that the integral is equal to 2pi/3.
