Chapter 9.4, Problem 45E

a.

To determine

Toshow: that if $f$ is an even function and the necessary integrals exist, then

  ${\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)dx=}2{\displaystyle {\int }_0^{\infty }f\left(x\right)dx}$

Explanation of Solution

Given information:

The function is an even function.

Proof: first break the limits of the integrals,

  ${\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)dx=}{\displaystyle {\int }_{-\infty }^{0}f\left(x\right)}dx+{\displaystyle {\int }_0^{\infty }f\left(x\right)dx}$

Now, since the function is an even function so substitute $x=-x$ , and corresponding limits,

  $\begin{array}{l}{\displaystyle {\int }_{-\infty }^{0}f\left(x\right)dx}={\displaystyle {\int }_0^{\infty }f\left(-x\right)}dx\text{ [substitute }x=-x\text{ and corresponding limits]}\\ {\displaystyle {\int }_{-\infty }^{0}f\left(x\right)dx}={\displaystyle {\int }_0^{\infty }f\left(x\right)}dx\text{ [}f\left(x\right)=f\left(-x\right)]\end{array}$

Thus,

  ${\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)dx=}2{\displaystyle {\int }_0^{\infty }f\left(x\right)dx}$

b.

To determine

To show: that if $f$ is an odd function and the necessary integrals exist, then

  ${\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)dx=}0$

Explanation of Solution

Given information:

The function is an even function.

Proof: first break the limits of the integrals,

  ${\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)dx=}{\displaystyle {\int }_{-\infty }^{0}f\left(x\right)}dx+{\displaystyle {\int }_0^{\infty }f\left(x\right)dx}$

Now, since the function is an even function so substitute $x=-x$ , and corresponding limits,

  $\begin{array}{l}{\displaystyle {\int }_{-\infty }^{0}f\left(x\right)dx}={\displaystyle {\int }_0^{\infty }f\left(-x\right)}dx\text{ [substitute }x=-x\text{ and corresponding limits]}\\ {\displaystyle {\int }_{-\infty }^{0}f\left(x\right)dx}=-{\displaystyle {\int }_0^{\infty }f\left(x\right)}dx\text{ [}f\left(-x\right)=-f\left(x\right)]\end{array}$

Thus,

  ${\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)dx=}0$