Dirichlet distribution - moments

[Boot20]

For a Dirichlet variable, I know the means and covariances, that is,

$E[X_i] = \alpha_i/\alpha_0$
$Cov[X_i,X_j] = \frac{ \alpha_i (\alpha_0I[i=j] - \alpha_j)}{\alpha_0^2(\alpha_0 + 1)}$

But how can I prove these facts?

 

[Boot20]

$p(x_1,...,x_{k-1}) = \frac{1}{B(\alpha) } \left[ \prod^{K}_{i=1} x_i^{\alpha_i - 1} \right]$

$E[X_1] = \frac{1}{B(\alpha) } \int^1_0 ...\int^{1 - \sum^{K}_{i=2}x_i}_0 x_1 \left[\prod^{K}_{i=1} x_i^{\alpha_i - 1} \right] d x_1 ... d x_{k}$

 

[bpet]

$p(x_1,...,x_{k-1}) = \frac{1}{B(\alpha) } \left[ \prod^{K}_{i=1} x_i^{\alpha_i - 1} \right]$

$E[X_1] = \frac{1}{B(\alpha) } \int^1_0 ...\int^{1 - \sum^{K}_{i=2}x_i}_0 x_1 \left[\prod^{K}_{i=1} x_i^{\alpha_i - 1} \right] d x_1 ... d x_{k}$

Hint: the expectation integrals closely resemble the definition of the normalizing constant B (which itself can be expressed in terms of gamma functions).