Given $\displaystyle (X_1,X_2,...,X_{k-1})$ has a Dirichlet distribution wiht parameters $\displaystyle (\alpha_1,...,\alpha_k)$

With $\displaystyle \alpha_{i+}=\alpha_i+...+\alpha_k$

I want to find $\displaystyle E[X_2]$

I know that $\displaystyle E[X_1]=\frac{\alpha_1}{\alpha_{1+}}$ since $\displaystyle X_1 \sim Beta(\alpha,\alpha_{2+})$

and

$\displaystyle (\frac{X_2}{1-X_1}|X_1) \sim Beta(\alpha_2,\alpha_{3+})$