Improper integral of $\sin^2(x)/x^2$ evaluated via residues I have come across another improper integral I wish to evaluate via residues. The integral is: $$\int_{-\infty}^\infty{\frac{\sin(x)^2}{x^2}}dx$$ $\sin(z)$ behaves in an uneasy way so I tried using the function $\frac{{e^{iz}}^2}{z^2}$ with a half circle on the upper complex plane with radius R and a half-circle of radius 1/R which arcs below $0$. The problem is the small semi-circles integral does not go to $0$ and in fact doesn't exist. What other types of contours or function substitutions should be used here?

Note that $ \cos(2x)=1-2\sin(x)^2 $, this suggest to consider the integral

$$ \int_{C} \frac{ {\rm e}^{2 i z} - 1 }{ z^2} dz \,.$$