Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyIndefinite IntegralsDefinition: Indefinite IntegralNotation for Indefinite IntegralsEvaluating Indefinite IntegralsRewrite the Indefinite IntegralInvestigation 03Investigation 04Use Technology to Rewrite Using Algebra and TrigonometryVerify AntiderivativesHomeworkCC 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.

An **Indefinite Integral** has the form:

The result when evaluating an Indefinite Integral is *if and only if*

**Details of the Definition of Indefinite Integrals**:

means that${F}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=f(x)$ is a$f(x)$ **derivative**function. means$\int f(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ *adding*values of the**derivative**function .$f(x)$ We will describe this in more detail soon.

The

*sum*of values results in the$f(x){\textstyle \phantom{\rule{0.167em}{0ex}}}\text{d}x$ function.$F(x)$ is called the$F(x)+C$ **antiderivative**of .$f(x)$

**NOTE**:

The phrase *if and only if* identifies the statement is a bi-conditional that means:

If

is true, then$\int f(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx=F(x)$ is also true;${F}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=f(x)$ **and**If

is true, then${F}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=f(x)$ is also true;$\int f(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx=F(x)$

The phrase *if and only if* is sometimes abbreviated *iff*.

The mathematical representation of the phrase *if and only if* is the symbol

*sums*, i.e., indefinite integrals, using continuous functions whereas *sums* using discrete functions (we will see this later in the course).

**NOTE**: Keep in mind, there is **NO** product rule and **NO** quotient rule for antiderivatives.

If you are not able to use your knowledge of derivatives in order to evaluate an indefinite integral, then you may need to rewrite the integral using algebraic or trigonometric properties.

Find the general indefinite integral by first re-writing, if needed, the integrand using the algebraic properties (see *PRE_01i_FTC_Algebra.pdf*) and trigonometric properties (see *PRE_01ii_FTC_Trig.pdf*).

Use FTC-I or technology to **verify** your answer.

$\int ({x}^{2}+\sqrt[3]{x}){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int ({x}^{2}-2x+1)(1-3x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \sqrt[3]{x}(1+x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{3{x}^{2}+5x-2}{x}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{(x+1{)}^{3}}{x}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{x-3\sqrt{x}+1}{\sqrt{x}}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{{x}^{2}-x+1}{{x}^{2}}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{1}{{x}^{2}+1}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int ({x}^{-3/4}+{x}^{1/4}-{e}^{x}){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{-1}{\sqrt{1-{x}^{2}}}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{\mathrm{sin}(x)}{{\mathrm{cos}}^{2}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{\mathrm{sin}(2x)}{\mathrm{cos}(x)}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{\mathrm{sin}(2x)}{\mathrm{sin}(x)}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int \frac{\mathrm{sec}(x)}{\mathrm{cos}(x)}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int [{\mathrm{cos}}^{2}{\textstyle \phantom{\rule{-0.167em}{0ex}}}\left(\frac{x}{2}\right)-{\mathrm{sin}}^{2}{\textstyle \phantom{\rule{-0.167em}{0ex}}}\left(\frac{x}{2}\right)]{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ $\int {[{\mathrm{sin}}^{2}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)+{\mathrm{cos}}^{2}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)]}^{3}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$

Explain why you need to first rewrite the integrals from **Investigation 03**.

**Mathematica**

**NOTE**: *Mathematica* may used on any Assessment to rewrite an expression without showing any work.

`x1``(* Simplify: (x^2-x+2)/x *)`

2`Simplify[(x^2-x+2)/x]`

3```
```

4`(* Simplify: (x^2-x+2)/x *)`

5`FullSimplify[(x^2-x+2)/x]`

6```
```

7`(* Expand: (x^2-x+2)(x+1) *)`

8`Expand[(x^2-x+2)(x+1)]`

9```
```

10`(* Rewrite: sin(x)/(cos(x))^2 *)`

11`TrigExpand[Sin[x]/Cos[x]^2]`

12```
```

13`(* Rewrite: (cos(x))^2 *)`

14`TrigReduce[Cos[x]^2]`

**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.$\text{E}{\textstyle \phantom{\rule{-0.167em}{0ex}}}\wedge {\textstyle \phantom{\rule{-0.167em}{0ex}}}2$ Do

**NOT**memorize the specifics of each of the above commands; know the commands and try them until the result is what you need.All of the commands shown above have only one (1) argument:

The expression to be simplified, if possible.

For help on using the above commands, e.g.,

*FullSimplify[ ]*function:In

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

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

Read how to use the

*FullSimplify[ ]*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***may**return a result in a*different,*that is**but equivalent**, form**unfamiliar**to you. In these cases, use another*Mathematica*command to rewrite the expression.

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.

`xxxxxxxxxx`

41`(* 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${\mathrm{sec}}^{2}(x)$ , is${e}^{x}$ , etc.*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.$F(x)$

At this time, you should be able to complete the following assignments:

**Section 4.9**[Review from Calculus I]: # 1 – 23 (odd);**Section 5.4**: # 1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

**Created**: Tuesday, 25 August 2020 03:24 EDT
**Last Modified**: Saturday, 28 May 2022 - 16:33 (EDT)