Assume: $$ \phi = \int f(x)p(x)dx = E_p(f)$$ Let $x_s \sim p, s=1,.....,S$ iid $(p(x_s=x)=p(x)$ and $p(x_1,x_2) = p(x_1) p(x_2)$. \begin{align} \hat{\phi} &= \frac{1}{S}\sum_{s=1}^{S}f(x_s) \\ E[\hat{\phi}] &= \frac{1}{S} \sum_{s=1}^{S}\int f(x_s) p(x_s) dx_s \\ &= \frac{1}{S} \sum_{s=1}^{S}E(f(x_s)) = \phi \end{align} So, we can call $\hat{\phi}$ as the unbiased estimator of $\phi$.
But how can $E(f(x_1)) = E(f(x_2))$...$=E(f(x_S))$ all be equal in order to get $\frac{1}{S} \sum_{s=1}^{S}E(f(x_s)) = \phi$? I think $p(x_1) = p(x_2)$...$=p(x_S)$ in this case but how can it be always the same?