Chapter 8.4, Problem 60E

a.

To determine

To identify: The value of $f\left(x\right)={\displaystyle \underset{0}{\overset{x}{\int }}\frac{\sin t}{t}dt}$ when $x\to \infty$ by investigating graphs and tables.

Answer to Problem 60E

The value of the function approaches to 1.57 approximately.

Explanation of Solution

Given information:

The given integral is $f\left(x\right)={\displaystyle \underset{0}{\overset{x}{\int }}\frac{\sin t}{t}dt}$ .

Consider the given function.

Draw the graph of the function on a grapher, plot $\text{NINT}\left(\frac{\sin x}{x},x,0,x\right)$ .

  

Here it can be observed that the graph is symmetrical about the y -axis, has an oscillating motion.

Observe the table as shown below:

  

From table and graph it can be observed that as x tends to infinity the limit approaches to 1.57 approximately.

b.

To determine

To identify: That the integral ${\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{\sin x}{x}dx}$ converges.

Answer to Problem 60E

The integral is an infinite sum of finite areas that alternate in sign, decrease in absolute value, and tends to a limit to zero.

The partial sum therefore close in on some limit.

Explanation of Solution

Given information:

The given integral is ${\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{\sin x}{x}dx}$ .

Consider the given integral.

The integral is an infinite sum of finite areas that alternate in sign, decrease in absolute value, and tends to a limit to zero.

The partial sum therefore close in on some limit.

  $\begin{array}{c}{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{\sin x}{x}dx}=\underset{x\to \infty }{\lim }{\displaystyle \underset{0}{\overset{x}{\int }}\frac{\sin x}{x}dx}\\ =\frac{\pi }{2}\end{array}$