Let $n\geq 2$ be an integer, and $$Q:=\left\{x=(x_1,\cdots,x_n)\in\mathbb{R}^n:\sum_{j=1}^{n}x_j=1,x_j\geq 0,j=1,\cdots,n\right\}$$ is the ($n-1$)-simplex in $\mathbb{R}^n$. Consider the map
$$ \Gamma:\quad Q\times (0,1)\rightarrow (0,1)^n $$
$$ \qquad\qquad\qquad\qquad \qquad(\beta,t)\mapsto t^{\beta}=(t^{\beta_1},\cdots,t^{\beta_n}). $$
I wonder if the formula $$ \int_{(0,1)^n}g(x)\,dx=\int_{Q}\int_{0}^1g(t^{\beta})|\log t|^{n-1}\,dt\,d\beta $$ is true or not.
And how to explain the appearance of $|\log t|^{n-1}$?
PS: I think it maybe from the Jacobian of the map $(x_1,\cdots,x_n)=(t^{\beta_1},\cdots,t^{\beta_n})$. And maybe coarea formula is useful.