I'm really stuck with this problem.
Is $X_t$ or $X_t-\mathbb{E}(X_t)$ a martingale, where $$ X_t=42+t^2W_t^3+tW_t^2+\int_0^t3W_udu+\int_0^tW_u^3dW_u ? $$
There is a hint, to find first $dX_t$, and then it says the solution will be very short.
Using Ito formula I find: $$ d(42)=0 $$ $$ d(t^2W_t^3)=2tW_t^3dt+3t^2W_t^2dW_t+3t^2W_tdt $$ $$ d(tW_t^2)=W_t^2dt+2tW_tdW_t+2tdt $$ $$ d(\int_0^t3W_udu)=3W_tdt $$ $$ d(\int_0^tW_u^3dW_u)=W_t^3dW_t $$ (the last one is a bit tricky. First, using Ito formula we find $$ d(W_t^4)=4W_t^3dW_t+6W_t^2dt, $$ and thus $$ \int_0^tW_u^3dW_u=\frac14W_t^4-\frac32\int_0^tW_u^2du. $$ The rest is straightforward application of Ito formula.)
Then I combine everything and find $$ d(X_t)=(2tW_t^3+3t^2W_t+W_t^2+2t+3W_t)dt+(3t^2W_t^2+2tW_t+W_t^3)dW_t. $$ Also after some calculations I find $$ \mathbb{E}(X_t)=42+t^2 $$
How is one supposed to infer from this whether $X_t$ or $X_t-\mathbb{E}(X_t)$ a martingale I have no idea...
I tried to solve it directly, using increments of the Wiener process, but the problem gets so messy so quick, that it is obvious there has to be a better way to do this.