Conditional expectation of integral

Question

$$E\Big(\int_0^2 t^2W_t^3 \, dt \mid F_1\Big)=\int_0^1 t^2W_t^3 \, dt +\int_1^2 E(t^2W_t^3 \mid F_1) \, dt=$$

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$E(W_t^3\mid F_1)=E((W_t-W_1+W_1)^3\mid F_1)=E((W_t-W_1)^3\mid F_1)+3E((W_t-W_1)^2W_1\mid F_1)+3E((W_t-W_1)W_1^2\mid F_1)+E(W_1^3\mid F_1)=0+3(t-1)W_1+0+W_1^3$

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$$=\int_0^1 t^2W_t^3\,dt +\int_1^2 t^23(t-1)W_1\,dt+\int_1^2 t^2 W_1^3\,dt=\int_0^1 t^2W_t^3\,dt+\frac{17}{4}W_1+\frac{7}{3}W_1^3$$

I would like to ask you if thats correct?

Answer 1

Note that $$\int_1^2 t^2 \, dt = \frac{7}{3} \neq \frac{1}{3}$$ (last step). Except for that minor mistake your calculations are correct.