This might be useful:
fSine[x_] := HeavisidePi[x/2]/(\[Pi]*Sqrt[1 - x^2]);
Chop@N[Integrate[fSine[x]*fSine[1/10 - x], {x, -2, 2}], 20]
and
N[Quiet[NIntegrate[fSine[x]*fSine[1/10 - x], {x, -2, 2}, WorkingPrecision -> 45, MaxRecursion -> 1000]], 20]
both give:
0.44420658153737745178
These work both with the Heaviside definition. We can achieve something similar with the original Piecewise definition.
fSine[x_] :=
Piecewise[{{1/(\[Pi]*Sqrt[1 - x^2]), -1 < x < 1}, {0, True}}];
Integrate[fSine[x]*fSine[1/10 - x], {x, -2, 2},
GenerateConditions -> False, PrincipalValue -> True];
Chop@N[%, 20]
which gives:
0.44420658153737745178
,too.
Cheers,
Marco
PS: Oh, yes, and
SetPrecision[N[Simplify@Integrate[fSine[x]*fSine[1/10 - x], {x, -2, 2}, GenerateConditions -> False, PrincipalValue -> True], 60], Infinity]
suggests that the imaginary part is really zero...