Show that:
$$\int_K \eta_0^{\alpha_0}(x)\cdots\eta_n(x)^{\alpha_n}dx=\frac{{\alpha_0}!\cdots{\alpha_n}!n!|K|}{(\alpha_0+\cdots+\alpha_n+n)!}$$
where $\eta_i$ barycentric coordinates and $K$ is a $n$-dim simplex.
I know that
$$\int_\tilde K \eta_0^{\alpha_0}(x)\cdots\eta_n(x)^{\alpha_n}dx=\frac{{\alpha_0}!\cdots{\alpha_n}!}{(\alpha_0+\cdots+\alpha_n+n)!}$$
where $\tilde K$ is the standard simplex. How do I get the factor $n!|K|$ from the transformation?
edit: If T is the trafo then $|det T|=n!|K|$