Integrate a function of an interpolating function

Question

I'm trying to plot the integral of an interpolating function multiplied by another (analytic) function, eg:
    sol = NDSolve[{ x''[t] == 1.0, x'[0] == 0, x[0] == 0}, {x'[t], 
   x[t]}, {t, 0, 1}, Method -> {"ImplicitRungeKutta", "DifferenceOrder" -> 4}]

then form some integral
f[t_] = Integrate[ t x'[t] /. sol, t]

and try to plot it
Plot[f[t], {t, 0, 1}]

This works fine if the integrand doesn't contain an analytic function (t, in the example above), but not if it does.

cvgmt · Accepted Answer

One way is use NIntegrate
sol = NDSolve[{x''[t] == 1.0, x'[0] == 0, x[0] == 0}, {x'[t], 
   x[t]}, {t, 0, 1}, 
  Method -> {"ImplicitRungeKutta", "DifferenceOrder" -> 4}]
f[s_?NumericQ] := NIntegrate[t*x'[t] /. sol, {t, 0, s}];
 Plot[f[s], {s, 0, 1}]

Another way is add y'[t]=t*x'[t].
sol2= NDSolve[{x''[t] == 1.0, x'[0] == 0, x[0] == 0, y'[t] == t*x'[t],
     y[0] == 0}, {y[t], x'[t], x[t]}, {t, 0, 1}, 
   Method -> {"ImplicitRungeKutta", "DifferenceOrder" -> 4}];
Plot[y[t] /. sol2, {t, 0, 1}]

Integrate a function of an interpolating function

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