Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyAntiderivativeDefinition: AntiderivativeFinding AntiderivativesActivity 01Activity 01, continuedInvestigation 02Use Technology to Verify AntiderivativesCC BY-NC-SA 4.0

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 4.9**– Antiderivatives**Section 5.3**– The Fundamental Theorem of Calculus**Section 5.4**– Indefinite Integrals and the Net Change Theorem

**Objective 01–01**: I can state the*Fundamental Theorem of Calculus*.**Objective 01–02**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–03**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–04**: I can determine the properties of an area function using*FTC-I*and*Calculus I*.**Objective 01–05**: I can evaluate indefinite integrals using the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–06**: I can evaluate definite integrals using the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–07**: I can use the*Net Change Theorem*to identify the meaning of .${\int}_{a}^{b}{f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}\text{d}x$ **Objective 01–08**: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.**Objective 01–09**: Given acceleration of an object in motion along a line, I can find the object’s velocity, displacement, and the total distance traveled.

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Recall from *Calculus I*,

An **Antiderivative** of

First, we will need to use our knowledge of derivatives:

**Example 01**:

Find the antiderivative of

**Solution**:

We need to find

From *Calculus I*, we know:

$\frac{d}{dx}(\mathrm{sec}(x)+1)=\mathrm{sec}(x)\mathrm{tan}(x)$ $\frac{d}{dx}\left(\textcolor[rgb]{}{\mathrm{sec}(x)-3}\right)=\mathrm{sec}(x)\mathrm{tan}(x)$ $\frac{d}{dx}\left(\textcolor[rgb]{}{\mathrm{sec}(x)+\pi}\right)=\mathrm{sec}(x)\mathrm{tan}(x)$

Thus, the antiderivative of

**NOTE**: *Python* is used here to quickly find a pattern.

**Activity**: Change the value of

**Warnings**:

Be very

**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Python*syntax is located at: https://www.jjw3.com/Common_Python_Code.html. lines are comment codes to explain the following executable code line(s).$\mathrm{\#}$ To execute code, press the “View the result” button:

What do you notice about the derivative,

Use your knowledge of derivatives (See *PRE_01iii_FTC_Calc_Rules.html*) and to find the following antiderivatives:

What is the antiderivative of

? Explain.$f(x)={e}^{x}$ What is the antiderivative of

? Explain.$f(x)=\mathrm{cos}(x)$ What is the antiderivative of

, i.e., evaluate? Explain.$f(x)={\mathrm{sec}}^{2}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ What is the antiderivative of

? Explain.$f(x)=-\mathrm{sin}(x)$ What is the antiderivative of

? Explain.$f(x)=\frac{1}{x}$ What is the antiderivative of

? Explain.$f(x)=\frac{1}{{x}^{2}+1}$

There are four (4) methods to check your work when finding antiderivatives.

Recall, **Example 01**: *find the antiderivative of*

**Method 01**: To check the antiderivative in **Example 01**,

Since

**Method 02**: Use **Mathematica**

**NOTE**: *Mathematica* is used here to **verify** antiderivatives. You must show that you understand the *Calculus I* concept of the antiderivative.

`x1``(* Example 01 *)`

2`(* Find antiderivative of sec(x)tan(x) *)`

3```
```

4`Integrate[Sec[x]Tan[x], x]`

**Warnings**:

Be very

**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Mathematica*syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.The

*Mathematica*syntax for is$\mathrm{tan}(x)$ ,$\text{Tan[x]}$ is${\mathrm{sec}}^{2}(x)$ ,$\text{Sec[x]}{\textstyle \phantom{\rule{-0.167em}{0ex}}}\wedge {\textstyle \phantom{\rule{-0.167em}{0ex}}}2$ is${e}^{x}$ , etc.$\text{E}{\textstyle \phantom{\rule{-0.167em}{0ex}}}\wedge {\textstyle \phantom{\rule{-0.167em}{0ex}}}2$ *Integrate[ ]*has at least two arguments:The integrand function;

The independent variable.

For help on using the

*Integrate[ ]*function:In

*Mathematica*, execute the code:$\text{?Integrate}$ Click on

near the bottom-left of output$\vee $ Click on local

Read how to use the

*Integrate[ ]*function – you will be able to copy-paste code.

Remember to use parens

and$($ to group multiple terms in the numerator or denominator of a rational expression.$)$ Remember, correct

*Mathematica*code will be all black except for variables.To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

*Mathematica*does**not**show for antiderivatives. You will need to remember to include$+C$ in your answers.$+C$ *Mathematica***may**return a result in a*different,*than your answer. In these cases, use another method to check your work.**but equivalent**, form

**Method 03**: Use **Python**

**NOTE**: *Python* is used here to **verify** antiderivatives. You must show that you understand the *Calculus I* concept of the antiderivative.

**Warnings**:

Be very

**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Python*syntax is located at: https://www.jjw3.com/Common_Python_Code.html.The

*Python*syntax for is${\mathrm{sec}}^{2}(x)$ ,$\text{sec(x)}{\textstyle \phantom{\rule{-0.167em}{0ex}}}\ast {\textstyle \phantom{\rule{-0.167em}{0ex}}}\ast 2$ is${e}^{x}$ ,$\text{exp(x)}$ is${x}^{4}$ , etc.$\text{x**4}$ *Python*requires for explicit multiplication, e.g.,$\ast $ is$\mathrm{sec}(x)\mathrm{tan}(x)$ .$\text{sec(x)*tan(x)}$ Remember to use parens

and$($ to group multiple terms in the numerator or denominator of a rational expression.$)$ *Python*does**not**show for antiderivatives. You will need to remember to include$+C$ in your answers.$+C$ *Python***may**return a result in a*different,*than your answer. In these cases, use another to check your work.**but equivalent**, formTo execute code, press the “View the result” button:

**Method 04**: Graph in **DESMOS**

**Warnings**:

In

*Box 1*, type the integrand function, . If you use function notation, use a lowercase$f(x)$ .$f$ In

*Box 2*, type in the antiderivative, . Make sure that you:$F(x)$ use function notation;

use an uppercase

;$F$ include

[this will create the slider in$+C$ *Box 3*];click on the color circle in

*Box 3*to hide the graph of .$F(x)$

In

*Box 4*, type .${\text{F}}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}\text{(x)}$ Compare the graphs of

and$f(x)$ . If the graphs are identical, then${F}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ , i.e.,${F}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=f(x)$ is the correct antiderivative of$F(x)$ .$f(x)$

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**Created**: Tuesday, 25 August 2020 03:24 EDT
**Last Modified**: Saturday, 28 May 2022 - 16:32 (EDT)