Vector Calculus | Part IA, 2005

Express the integral

I=0dx01dy0xdzxeAx/yBxyCyzI=\int_{0}^{\infty} d x \int_{0}^{1} d y \int_{0}^{x} d z x e^{-A x / y-B x y-C y z}

in terms of the new variables α=x/y,β=xy\alpha=x / y, \beta=x y, and γ=yz\gamma=y z. Hence show that

I=12A(A+B)(A+B+C)I=\frac{1}{2 A(A+B)(A+B+C)}

You may assume A,BA, B and CC are positive. [Hint: Remember to calculate the limits of the integral.]

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