ok. Let's denote values {t, f(t)} as F

Then interpolate this array with function ListInterpolation.

fx = ListInterpolation[F[[All, 2]], {F[[1,1]],F[[-1, 1]]}]

Then you may use Integrate with fx

Integrate[Sin[2*Pi*t]*fx[t], {x,F[[1,1]],F[[-1, 1]]}]

As you see it's not exactly what you want. Measures of integlal is the {F[[1,1]], F[[-1, 1]]}.

Let's denote values {t, f(t)} as F, then interpolate this array with ListInterpolation.

fx = ListInterpolation[F[[All, 2]], {F[[1, 1]], F[[-1, 1]]}]

Now we may use Integrate with fx

Integrate[Sin[2*Pi*t]*fx[t], {x, F[[1, 1]], F[[-1, 1]]}]

As you see, it's not exactly what you want: the domain of integration is {F[[1, 1]], F[[-1, 1]]}.