r1 = Range[-10,0.99,0.01];

r2 = Range[1.01,10,0.01];

f11 := (Sqrt[2π]Sin[π (x-1)]/(π (x-1)))/. x->r1;

f12 := (Sqrt[2π]Sin[π (x-1)]/(π (x-1)))/. x->r2;

f21 := (Integrate[1/Sqrt[2π](Sin[ω x]Sin[ω ]+ Cos[ω x]Cos[ω ]),{ω,-π,π}])/. x->r1;

f22 := (Integrate[1/Sqrt[2π](Sin[ω x]Sin[ω ]+ Cos[ω x]Cos[ω ]),{ω,-π,π}])/. x->r2;

d1 = f11 - f21;

d2 = f12 - f22;

Max[Abs[d1]]

Max[Abs[d2]]

Plot[(Integrate[1/Sqrt[2π](Sin[ω x]Sin[ω ]+ Cos[ω x]Cos[ω ]),{ω,-π,π}]), {x, -10,10}]

Plot[(Sqrt[2π]Sin[π (x-1)]/(π (x-1))), {x, -10,10}]