Chapter 6.2, Problem 78E

$\text{(a)}$

To determine

To explain: How the derivative of $F(x)+C$ can be used to verify the integration is accurate.

Explanation of Solution

Given information:

The integral is ${{\displaystyle \int }}^{\text{}}f(x)dx=F(x)+C$

Explain:

If the derivative of $F(x)+C$ is equal to $f(x)$ then the integration was performed correctly.

$\text{(b)}$

To determine

To explain: How use the slope field for $dy/dx=f(x)$ and the graph of $y=F(x)$ to support evaluation of the integral.

Explanation of Solution

Given information:

The integral is ${{\displaystyle \int }}^{\text{}}f(x)dx=F(x)+C$

Explain:

The slope field generated by $f(x)$ should generate all curves in the family $y=F(x)+C$ .

$(c)$

To determine

To explain: How use the graph of $C$ . $y_1=F(x)$ and to support evaluation of the integral.

Explanation of Solution

Given information:

The integral is ${{\displaystyle \int }}^{\text{}}f(x)dx=F(x)+C$

Explain:

  $y_2$ evaluates to $F(x)-F(0)$ ,

So,

are the same graph except one is a vertical shift of the other.

$(d)$

To determine

To explain: How can use the tables of values $y_1-y_2,y_1\text{, }{y}_2$ defined as in part $(c)$ to support evaluation of the integral.

Explanation of Solution

Given information:

The integral is ${{\displaystyle \int }}^{\text{}}f(x)dx=F(x)+C$

Explain:

Evaluate $y_1-y_2$ at various places using the values listed in the table. If the integration was done correctly, the outcome will be constant for all locations.

$(e)$

To determine

To explain: How can use the graph of $f$ and NDER of $F(x)$ to support evaluation of the integral.

Explanation of Solution

Given information:

The integral is ${{\displaystyle \int }}^{\text{}}f(x)dx=F(x)+C$

Explain:

The graph obtained use NDER of $F(x)$ should correspond exactly to the graph $y=f(x)$ .

$(f)$

To determine

To find: The value of part $(a)-(e)$ .

Answer to Problem 78E

The answer: $F(x)=\sqrt{x^2+1}+C$

Explanation of Solution

Given information:

The integral is ${{\displaystyle \int }}^{\text{}}f(x)dx=F(x)+C$

Calculation:

  $\begin{array}{c}u=x^2+1\\ du=2xdx\\ {{\displaystyle \int }}^{\text{}}\frac{x}{\sqrt{x^2+1}}dx\\ =\frac{1}{2}{{\displaystyle \int }}^{\text{}}\frac{du}{\sqrt{u}}\\ =\sqrt{u}+C\\ =\sqrt{x^2+1}+C\end{array}$

Therefore, the required value is $F(x)=\sqrt{x^2+1}+C$ .