I was wondering about the statistical variance of $A\sin(F x)+ Bx$, and the covariance (or correlation) between $A_{1} \sin(F_{1} x) + B_{1} x$ and $A_{2} \sin(F_{2} x) + B_{2} x$, assuming that $x$ is uniformly distributed in $[0, U]$, and $A, A_1, A_2, F, F_1, F_2, B, U$ are constants.
How can I proceed to compute these quantities in the simplest way?
I'm going to simplify your notation some. I will have capital letters refer to random variables and lowercase letters refer to the parameters. To that end, suppose $X \sim \mathcal{U}(0, u)$ (that is, $X$ follows a uniform distribution on $[0, u]$). Consider the random variable $Y = a\sin(\lambda X) + bX.$ I'll leave it to you to compute the variance, knowing that $$\sigma_X ^2 = E(X^2) - (E(X))^2$$
(hint: integrate by parts). Once you have the variance, now use the correlation formula $$\rho_{X, Y} = \frac{\mathrm{Cov}(X, Y)}{\sigma_X \sigma_Y}$$
where the covariance is a double integral in this case, i.e.
$$ \mathrm{Cov}(X, Y) =E\left[ (X - E(X))(Y - E(Y) \right].$$
This, again, may require parts.