As part of an exercise I am doing from stochastic analysis I need to compute the following integral: $$E(\int_0^x \mathrm{sin}(X_t+Y_t) \mathrm{cos}(X_t+Y_t)dt)$$ where $X_0=0$ and $Y_0=0$. Presumably it is equal to $0$ but I don't know how to show it.
Edit: added the expected value
We have: $$d(X_tY_t)=X_tdY_t + Y_tdX_t+d\langle X,Y\rangle_t={X_t\cos(X_t+Y_t)dW_t}+{Y_t\sin(X_t+Y_t)dW_t}+{\sin(X_t+Y_t)\cos(X_t+Y_t)dt}$$
Can you take it from here or do you need me to continue?
EDIT: the two Brownian motions are in fact independent so the last term is 0. As said in a comment, you have to proceed differently.