# Fourth moment of a itos integral

$$I(t)=\int_0^t \sqrt sdW_s$$

What is $$E(I(t)^4)$$

$$I(t)=\int_0^t \sqrt tdW_s=\sqrt t \int_0^t dW_s =\sqrt t W_t$$ and then $$E(I(t)^4)=E(t^2 W_t^4)=t^2 \cdot 3t^2=3t^4$$ using the 4th moment of the $$N(0,\sigma^2=t)$$ distribution.
Note that because the integrand is deterministic this Itô integral is normally distributed with parameters (cf. Itô isometry) $$I_t := \int_0^t \sqrt{s} dW_s \sim N(0, t^2/2)$$ Now you can just use the results that apply for the moments of a Gaussian variable.