Double integral of an exponential function .

Question

I need to integrate, $$I=\int_0^\infty\int_0^\infty \frac{e^{-(x+y)}}{x+y}\ dx\ dy$$

I tried using integration by parts for the first integrand but I am getting its value as zero.

Answer 1

Set $x = uv$, $y=u(1-v)$, which changes the bounds to $0<u<\infty$ and $0<v<1$ (this is a common trick for integration over the first quadrant). Then $dx \, dy = u \, du \, dv $, and $x+y = u $, so the integral becomes $$ \int_0^1 \int_0^{\infty} e^{-u} \, du \, dv, $$ which is easy.