I've been having trouble figuring out how to directly prove that the covariance of the multivariate Gaussian distribution $p(x) = \dfrac{1}{{{(2\pi)}^{\frac{n}{2}}}|\Sigma|^{\frac{1}{2}}}\exp\{{-\dfrac{1}{2}(x - \mu)^T \Sigma^{-1}(x-\mu)}\}$ where $x \in \mathbb{R}^n$ and $\Sigma$ is positive definite, is actually $\Sigma$.
That is to say, given $\Sigma$ and the above density function, I want to prove that the covariance is $\Sigma$. I've been able to prove that the mean is $\mu$ by diagonalizing the matrix $\Sigma$ and integrating, but I'm struggling to do it for the proof that the covariance is $\Sigma$. Can anyone help?