how to double integrate this exponential function

Question

how can i integrate this $\int_0^∞\int_x^∞{\frac{e^{-y}}{y}}dy dx$

i am stuck here $\int{\frac{e^{-y}}{y}}dy$. I have tried some log substitutions.

Answer 1

There is no elementary antiderivative of $e^{-y}/y$

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A much better approach, since the integrands are positive, is to change the order of integration: This leads to

$$\int_0^{\infty} \int_0^y \frac{e^{-y}}{y} dx dy = \int_0^{\infty} \frac{e^{-y}}{y} y dy$$