Stochastic integral involving Wiener process

Question

Suppose $W_t$ is the standard Wiener process. I am wondering whether I can use the independent increment property of the process as well as the fact that $W_t$ is zero-mean Gaussian to evaluate expectation of the following stochastic integral as I have: $$ \mathbb{E}\Big[ \int\limits_{0}^t W^3_s dW_s \Big] = \int\limits_{0}^t \mathbb{E}[W^3_s] \; \mathbb{E}[dW_s] \\ = \int\limits_{0}^t \underbrace{\mathbb{E}[W^3_s]}_{=0} \; \mathbb{E}[dW_s] \\ = 0.$$ I am wondering whether I have used the techniques soundly.