Chapter 16.2, Problem 99E

Improper integrals Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 8.9). For example, under suitable conditions on f,

${\displaystyle {\int }_a^{\infty }{\displaystyle {\int }_g^{h\left(x\right)}f\left(x,y\right)dydx=\underset{x\to \infty }{\lim }}}{\displaystyle {\int }_a^b{\displaystyle {\int }_{g\left(x\right)}^{h\left(x\right)}f\left(x,y\right)dydx}}$

Use or extend the one-variable methods for improper integrals to evaluate the following integrals.

99. ${\displaystyle {\int }_1^{\infty }{\displaystyle {\int }_0^{e^{-y}}xydydx}}$