Evaluating Integral of $x_1^{b_1-1}\cdots x_k^{b_k-1}$ over $x_1+\dots+x_k\leq 1$

I am struggling with this Integral: $$\idotsint\limits_{\begin{subarray}{l}x_1+\dots+x_k\leq 1 \\ x1,\dots,x_k \geq 0 \end {subarray}} x_1^{b_1-1}\cdots x_k^{b_k-1} dx_1 \cdots dx_k $$ for $b_1,\dots,b_k>0$.

My attempt so far is to use Fubini's theorem and get to $\frac{x_1^{b_1}\cdots x_k^{b_k}}{b_1\cdots b_k}$ but I am struggling to evaluate for $\Big|_{\begin{subarray}{l}x_1+\dots+x_k\leq 1 \\ x1,\dots,x_k \geq 0 \end {subarray}}$. I have also thought about using induction and while $k=1$ is clear, I'm having a hard time with $k\geq 2$.

Solution 1:

Try a change of variables to $y_1,\dots,y_n$ where $$ y_1=x_1\, ,\ x_2=(1-x_1)y_2\, ,\ x_3=(1-x_1-x_2)y_3\, , $$ up to $$ x_n\, =\, (1-(x_1+\dots+x_{n-1}))y_n\, . $$ Then your domain is $(y_1,\dots,y_n) \in [0,1]^n$. Note that $x_2=(1-y_1)y_2$ and $$ x_3\, =\, (1-y_1 - (1-y_1)y_2)y_3\, =\, (1-y_1)(1-y_2)y_3\, . $$ Then inductively, we have $x_k = (1-y_1)(1-y_2)\cdots (1-y_{k-1})y_k$ and $1-(x_1+\dots+x_k) = (1-y_1)(1-y_2)\cdots (1-y_{k-1})(1-y_k)$. Also, if $y_1,\dots,y_{k-1}$ are held fixed, then $$ dx_k\, =\, (1-y_1)(1-y_2)\cdots (1-y_{k-1})\, dy_k\, . $$ So you can turn your iterated integrals into a product of Beta-integrals which you can evaluate using $B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$.