Chapter 5, Problem 17RQ

To determine

To Find: whether $f$ is continuous, then ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x)dx=\underset{t\to \infty }{\lim }}{\displaystyle \underset{-t}{\overset{t}{\int }}f(x)dx}$ is true.

Answer to Problem 17RQ

When improper integrals going to infinity in both directions then, first separate that integral

into the sum of two integrals, one going from a set number to positive infinity and other

going from the same number to negative infinity

Explanation of Solution

Given information: $f\left(x\right)$ is continuous and ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x)dx=\underset{t\to \infty }{\lim }}{\displaystyle \underset{-t}{\overset{t}{\int }}f(x)dx}$

When dealing with improper integrals going to infinity in both directions, first separate that integral into the sum of two integrals, one going from a set number to positive infinity and other going from the same number to negative infinity. Then take the limit of each one seperatly

  ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x)dx}={\displaystyle \underset{-\infty }{\overset{a}{\int }}f(x)dx+{\displaystyle \underset{a}{\overset{\infty }{\int }}f(x)dx}}$

  $a$ can be any real number