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]]}`.