Given this random vector:
$$ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $$
And this probability distribution function that takes it as argument:
$$ f_\mathbf{X}(\mathbf{x}) = \begin{cases} 4x_1x_2e^{-(x_1^2+x_2^2)} & x_1,x_2 \ge 0 \\ 0 & \text{otherwise} \end{cases} $$
How do I calculate expected value of $\mathbf{x}$; that is $E\{\mathbf{x}\}$?
My aim is to calculate the covariance matrix
$$ \mathbf{C_x} = E\left\{(\mathbf{x}-E\{\mathbf{x}\})(\mathbf{x}-E\{\mathbf{x}\})^T\right\} $$
But I'm stuck at finding mean of $\mathbf{x}$ vector.
As noted by Stefan Hansen in comments, expected values can be found term-by-term.
Notice how your density factors: $\Big(2x_1 e^{-x_1^2}\Big)\Big(2x_2 e^{-x_2^2}\Big)$. Since it's a function of $x_1$ times a function of $x_2$, the two components of your random vector are independent, so the off-diagonal entries of the matrix will be $0$. And because of independence, you can find the expected values separately. And finally, because of symmetry, the two expected values are the same.