Chapter 8.9, Problem 92E

Improper integrals with infinite intervals and unbounded integrands For a real number a. suppose $\underset{x\to a^{+}}{\lim }f(x)=-\infty$ or $\underset{x\to a^{+}}{\lim }f(x)=\infty$. In these cases, the integral ${\displaystyle {\int }_a^{\infty }f(x)}$dx is improper for two reasons: ∞ appears in the upper limit and f is unbounded at x = a. It can be shown that ${\displaystyle {\int }_a^{\infty }f(x)}dx={\displaystyle {\int }_a^{c}f(x)}dx+{\displaystyle {\int }_c^{\infty }f(x)}dx$, for any c > a. Use this result to evaluate the following improper integrals.

92. ${\displaystyle {\int }_0^{\infty }{x}^{-x}(\ln x+1)dx}$