Chapter 8.7, Problem 86E

(a)

To determine

Show that the both integral terms ${\displaystyle \underset{-\infty }{\overset{a}{\int }}f\left(x\right)dx}$ and ${\displaystyle \underset{a}{\overset{\infty }{\int }}f\left(x\right)dx}$ converge if and only if the integral terms ${\displaystyle \underset{-\infty }{\overset{b}{\int }}f\left(x\right)dx}$ and ${\displaystyle \underset{b}{\overset{\infty }{\int }}f\left(x\right)dx}$ will converge.

(b)

To determine

Show that the sum of both integral terms ${\displaystyle \underset{-\infty }{\overset{a}{\int }}f\left(x\right)dx}$ and ${\displaystyle \underset{a}{\overset{\infty }{\int }}f\left(x\right)dx}$ is equal to the sum of the integral terms ${\displaystyle \underset{-\infty }{\overset{b}{\int }}f\left(x\right)dx}$ and ${\displaystyle \underset{b}{\overset{\infty }{\int }}f\left(x\right)dx}$, when integrals involved actually converges.