Expected value of product of an ito integral and a random variable

Question

I want to compute $$E[\int_0^t W_r dr \int_0^s W_r^2 dW_r].$$ Here $t,s$ are arbitrary. I have thought about this a lot but not sure how to proceed. I tried to apply Ito's formula to one of the factors in the product, but that did not seem to help. Any idea will be appreciated!

Answer 1

I'd start by using Ito's formula to write $$ \int_0^s W^2_r\,dW_r={1\over 3}W^3_s-\int_0^s W_u\,du. $$