Chapter 15: Multiple Integrals - Section 15.2 - Double Integrals over General Regions - Exercises 15.2 - Page 883: 69

$1$

Consider $I= \int_{1}^{\infty} \int_{e^{-x}}^{1} \dfrac{1}{x^3y} dy dx$ or, $= \int_{1}^{\infty}[\dfrac{\ln 1}{x^3}-\dfrac{\ln e^{-x}}{x^3}] dx$ or, $=\int_{1}^{\infty} 0-\dfrac{-x}{x^3} dx$ Thus, we have $I=[-x^{-1}]_1^{\infty} =0+1=1$