Chapter 6.4, Problem 71E

(a)

To determine

To prove: Si(x) is an odd function of x .

Explanation of Solution

Given information: 

  $f(t)=\{\begin{array}{l}\frac{\sin t}{t}\text{, t}\ne \text{0}\\ 1\text{ , t}=\text{0}\end{array}$

Formula used:

Odd function:

  $f(-x)=-f(x)$

Proof: 

  $\begin{array}{l}f(t)=\{\begin{array}{l}\frac{\sin t}{t}\text{, t}\ne \text{0}\\ 1\text{ , t}=\text{0}\end{array}\\ f(-t)=\{\begin{array}{l}\frac{\sin (-t)}{-t}\\ 1\end{array}\\ f(-t)=\{\begin{array}{l}-\frac{\sin (t)}{t}\\ 1\end{array}\\ f(-t)=-f(t)\end{array}$

So it is a odd function.

(b)

To determine

To calculate: Value of Si(0)

Answer to Problem 71E

  $Si\left(0\right)=0$

Explanation of Solution

Given information: $Si(x)={\displaystyle \int \underset{0}{\overset{x}{}}\frac{\sin t}{t}dt}$

Calculation:

  $\begin{array}{l}Si(x)={\displaystyle \int \underset{0}{\overset{x}{}}\frac{\sin t}{t}dt}\\ Si(0)={\displaystyle \int \underset{0}{\overset{0}{}}\frac{\sin t}{t}dt}\end{array}$

As limits of the function is same so answer of the function will be zero.

(c)

To determine

To calculate:

Local extremes of the function Si(x).

Answer to Problem 71E

Local extreme is x=0

Explanation of Solution

Given information:

  $Si(x)={\displaystyle \int \underset{0}{\overset{x}{}}\frac{\sin t}{t}dt}$

Formula used:

  ${\displaystyle \int f(x)g(x)dx=f(x){\displaystyle \int g(x)dx-{\displaystyle \int (f\text{'}(x){\displaystyle \int g(x)dx)dx}}}}$

Calculation:

  $Si(x)={\displaystyle \int \underset{0}{\overset{x}{}}\frac{\sin t}{t}dt}$

  $\begin{array}{l}{\displaystyle \int f(x)g(x)dx=f(x){\displaystyle \int g(x)dx-{\displaystyle \int (f\text{'}(x){\displaystyle \int g(x)dx)dx}}}}\\ {\displaystyle \int \frac{\mathrm{Sin}x}{x}dx=\frac{1}{x}}{\displaystyle \int \sin xdx-{\displaystyle \int (\cos x{\displaystyle \int \frac{1}{x}dx)dx}}}\\ -\sin x-{\displaystyle \int (\cos x\log x)dx}\\ -\sin x-[(\cos x(x\log x-x)-\log x(\sin x))]\\ -\sin x-x\cos x\log x+x\cos x+\sin x\log x=0\\ x\cos x-\sin x-(x\cos -\sin x)\log x=0\\ 1-\log x=0\\ \log x=1\\ x=0\end{array}$

(d)

To determine

To graph: Si(x)

Explanation of Solution

Given information:

  $Si(x)={\displaystyle \int \underset{0}{\overset{x}{}}\frac{\sin t}{t}dt}$

Graph: