NIntegrate[x^50*Sin[x], {x, 0, 1}] works fine and results to 0.0162898.

Also putting N[Integrate[x^50*Sin[x], {x, 0, 1}], 20] you get a correct result.

If you try to calculate the anti-derivative using :

anti[x_] := Module[{t}, Integrate[t^50*Sin[t], t] /. t -> x] then you will see that you get the exact integral form.

Now

anti[1]==16432804687774250383441481995831940788236063969597816674967907249 Cos[1] + 25592576958484906358990554067594971071796795711786890514692693650 Sin[1]

and

anti[0]==30414093201713378043612608166064768844377641568960512000000000000

N[anti[1] - anti[0], 20]==0.016289783620195801683 but N[anti[1]-anti[0]]==0. which is indeed strange to me also.

NIntegrate[x^50*Sin[x], {x, 0, 1}] works fine and results to 0.0162898.

Also putting N[Integrate[x^50*Sin[x], {x, 0, 1}], 20] you get a correct result.

If you try to calculate the anti-derivative using :

anti[x_] := Module[{t}, Integrate[t^50*Sin[t], t] /. t -> x] then you will see that you get the exact integral form.

Now

anti[1]==16432804687774250383441481995831940788236063969597816674967907249 Cos[1] + 25592576958484906358990554067594971071796795711786890514692693650 Sin[1]

and

anti[0]==30414093201713378043612608166064768844377641568960512000000000000

N[anti[1] - anti[0], 20]==0.016289783620195801683 but N[anti[1]-anti[0]]==0. which is caused by the round to only 8 digits of Sin[1] and Cos[1].

Try Trace[N[Integrate[x^50*Sin[x], {x, 0, 1}]]] to see where exactly it goes way off!