How do you calculate $E[x\sin(x)]$ if x is a uniform random variable between $-\pi/2$ and $\pi/2$?
Additionally, what is the relation of finding covariance of $XY$ to finding the W and b for Linear MMSE ($LMMSE$)? [I gather that $E[X]$ and $E[Y]$ are $0$ so for calculating covariance matrix of $XY$ ($\Sigma_{XY}$) I need to only calculate $E[x\sin(x)]$]
Since the probability denisty function for $X$ is $\tfrac 1\pi$ over the support $[-\pi/2;\pi/2]$ then it is a matter of integration
$$\mathsf E(X\sin X) ~{ =\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} x\sin x~\mathsf d x \\= \frac 1\pi\Big[\sin(x)-x\cos(x)\Big]_{x=-\pi/2}^{x=\pi/2} \\= \frac 2\pi}$$