$K$ is a baseline kernel function that is nonnegative, symmetric and supported on [-1,1]; $h$ is a bandwidth in local smoothing .Let $p_j$ denote the marginal density of $X_j$.We define the estimator of $p_j$ as $$\hat{p}_j(x_j)=n^{-1}\sum_{i=1}^{n} K_{h_j}(x_j,X_j^i)$$ ,where $K_h(x,u)=h^{-1}K(\dfrac{x-u}{h}) \text{ if } x \in [2h, 1-2h] \text{ or } u \in [h,1-h]$.
Also, there is an assumption that $\int_{-1}^{1}K(t)dt=1$.
I want to calculate $$\int_0^1 x_j \hat{p}_j(x_j) dx_j $$. Here, $X_j^i$ means the $i$th observation of $X_j$.
Any form is fine. But I want to know whether I can eliminate the integral sign and make it to a simpler form.