Feynman Parameters

$$ \int_0^\infty dy_1 \dots \int_0^\infty dy_n e^{-A_1 y_1-\cdots-A_n y_n} $$ $$ = \int_0^\infty dy_1 \dots \int_0^\infty dy_n e^{-A_1 y_1-\cdots-A_n y_n} \int_{s=0}^\infty ds \delta\left(\sum_{j=1}^n y_j - s\right) $$ $$ = \int_{s=0}^\infty ds \int_0^\infty dy_1 \dots \int_0^\infty dy_n e^{-A_1 y_1-\cdots-A_n y_n}\delta\left(\sum_{j=1}^n y_j - s\right) $$ Make the substitution $y_j = s x_j$, and note $\delta(sx) = s^{-1} \delta(x)$ if $s > 0$, to get $$ = \int_{s=0}^\infty ds \, s^{n-1} \int_0^\infty dx_1 \dots \int_0^\infty dx_n e^{-(A_1 x_1-\cdots-A_n x_n)s}\delta\left(\sum_{j=1}^n x_j - 1\right) $$ $$ = \int_0^\infty dx_1 \dots \int_0^\infty dx_n \delta\left(\sum_{j=1}^n x_j - 1\right) \int_{s=0}^\infty ds \, s^{n-1} e^{-(A_1 x_1-\cdots-A_n x_n)s} $$ and evaluate the integral over $s$ to get $$ = \int_0^\infty dx_1 \dots \int_0^\infty dx_n \delta\left(\sum_{j=1}^n x_j - 1\right) \frac{(n-1)!}{(\sum_{j=1}^nx_jA_j)^{n}} $$ But also $$ \int_0^\infty dy_1 \dots \int_0^\infty dy_n e^{-A_1 y_1-\cdots-A_n y_n} $$ $$ = \prod_{j=1}^n \int_0^\infty dy_j e^{-A_j y_j} = \left(\prod_{j=1}^n A_j \right)^{-1} .$$