The Ito integral of BM is 1/2(Wt^T + T) at the limit, as shown by for example Shreve in his online lecture notes.Can someone tell me what the Ito integral Wt^2 dwt in the limit is please - can I find it using a similar method? The only way I can find an answer is by applying Ito's lemma to 1/3Wt^3 and rearranging.Thanks for any pointers.

The method you suggest is the only practical way of solving this. You "guess" what the integral will look like, then apply Ito's lemma and rearrange.

..and as a complement to Tibbar's answer, it is not that surprising you should have to do this given it is the same procedure as has to be applied for non-trivial deterministic ODEs and PDEs.However, for the simple polynomial cases you mention, the guess can be highly educated indeed. The reason why you guess 1/3 (W_t)^3 in your example is that this would be the result in the deterministic case, and you know that the Ito result ends up as the deterministic result alongside a correction term due to the non-negligible square term in the expansion. Applying Ito's Lemma to the deterministic guess will make that correction term explicit.I hope this helps.