The value of the integral ∞∫-∞sin x dx isx²

The value of the integral
^∞∫_-∞ sin x dx is
x² + 2x + 2

evaluated using contour integration and the residue theorem is

I = ^∞∫_-∞	sin (x)
	x² + 2x + 2

let f(x) =	I_m(e^ix)
	z² + 2z + 2

Then poles of f(z) are given by z² + 2z + 2 = 0
∴ z = – 1 ± i
R₁ = Res: (f(z): z = -1 + i)

Lt_z→-1±1[z-(-1+i)]	e^ix
	[z-(-1+1)][z-(-1-i)]

=	e^i(-1+i)	=	e^-i-1
	-1 + i + 1 + i		2i

= I_M[πe^-1(cos(1) - isin(1))] = -	π sin(1)
	e