It is not efficient to compute derivatives n times if you already computed derivative n-1 NestList keeps track of the result, so with a little modification this can do the computation you need for both derivative and integration

Clear[f, x];
f[x_] = Sin[x];
TableForm[NestList[D[#, x] &, f[x], 5]]
TableForm[NestList[Integrate[#, x] &, f[x], 5]]

If you do not want the first element in the list (which is f[x], than just use Rest to drop the first element of the list, that is

Clear[f, x];
f[x_] = Sin[x];
TableForm[Rest@NestList[D[#, x] &, f[x], 5]]
TableForm[Rest@NestList[Integrate[#, x] &, f[x], 5]]

best yehuda