# Fundamental Theorem of Calculus (FTC)

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

### Expected Educational Results

• Objective 01–01: I can state the Fundamental Theorem of Calculus.

• Objective 01–02: I can explain the meaning of the Fundamental Theorem of CalculusPart I.

• Objective 01–03: I can explain the meaning of the Fundamental Theorem of CalculusPart 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 CalculusPart I.

• Objective 01–06: I can evaluate definite integrals using the Fundamental Theorem of CalculusPart II.

• Objective 01–07: I can use the Net Change Theorem to identify the meaning of ${\int }_{a}^{b}{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)\phantom{\rule{0.167em}{0ex}}\text{d}x$$\int_a^b{f^{\,\prime}(x)\,\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.

### Bloom’s Taxonomy

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.

## Antiderivative

### Definition: Antiderivative

Recall from Calculus I,

An Antiderivative of $f\left(x\right)$$f(x)$ is any function $F\left(x\right)$$F(x)$ whose derivative is ${F}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)=f\left(x\right)$$F^{\,\prime}(x) = f(x)$.

### Finding Antiderivatives

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

Example 01:

Find the antiderivative of $f\left(x\right)=\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)$$f(x)=\sec{(x)}\tan{(x)}$

Solution:

We need to find $F\left(x\right)$$F(x)$ so that $\frac{d}{dx}F\left(x\right)=f\left(x\right)$$\frac{d}{dx}F(x)=f(x)$.

From Calculus I, we know:

• $\frac{d}{dx}\left(\mathrm{sec}\left(x\right)+1\right)=\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)$$\frac{d}{dx}\left(\sec{(x)}+1\right) = \sec{(x)}\tan{(x)}$

• $\frac{d}{dx}\left(\mathrm{sec}\left(x\right)-3\right)=\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)$$\frac{d}{dx}\left({\color{gsured}\sec{(x)}-3}\right) = \sec{(x)}\tan{(x)}$

• $\frac{d}{dx}\left(\mathrm{sec}\left(x\right)+\pi \right)=\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)$$\frac{d}{dx}\left({\color{gsured}\sec{(x)}+\pi}\right) = \sec{(x)}\tan{(x)}$

Thus, the antiderivative of $f\left(x\right)=\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)$$f(x)=\sec{(x)}\tan{(x)}$ is $F\left(x\right)=\mathrm{sec}\left(x\right)+C$$F(x)=\sec{(x)}+C$, where $C$$C$ is any constant.

#### Activity 01

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

Activity: Change the value of $C$$C$ in line 8, then execute the code.

Warnings:

1. 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.

2. $\mathrm{#}$$\#$ lines are comment codes to explain the following executable code line(s).

3. To execute code, press the “View the result” button:

#### Activity 01, continued

What do you notice about the derivative, $\frac{d}{dx}\left(\mathrm{sec}\left(x\right)+C\right)$$\frac{d}{dx}\left(\sec{(x)}+C\right)$?

#### Investigation 02

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

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

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

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

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

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

6. What is the antiderivative of $f\left(x\right)=\frac{1}{{x}^{2}+1}$$f(x)=\frac{1}{x^2+1}$? Explain.

#### Use Technology to Verify Antiderivatives

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

Recall, Example 01: find the antiderivative of $f\left(x\right)=\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)$$f(x)=\color{green}\sec{(x)}\tan{(x)}$. The antiderivative was determined to be $F\left(x\right)=\mathrm{sec}\left(x\right)+C$$F(x)={\color{red}\sec{(x)} + C}$.

Method 01: To check the antiderivative in Example 01, $F\left(x\right)$$F(x)$, differentiate the antiderivative, i.e., $\frac{\text{d}}{\text{d}x}F\left(x\right)$$\frac{\text{d}}{\text{d}x}F(x)$, and compare the derivative to $f\left(x\right)$$f(x)$.

$\frac{\text{d}}{\text{d}x}F\left(x\right)=\frac{\text{d}}{\text{d}x}\left(\mathrm{sec}\left(x\right)+C\right)=\frac{\text{d}}{\text{d}x}\mathrm{sec}\left(x\right)+\frac{\text{d}}{\text{d}x}C=\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)+0=f\left(x\right)$$\displaystyle \frac{\text{d}}{\text{d}x}F(x)=\frac{\text{d}}{\text{d}x}\left({\color{red}\sec{(x)} + C}\right)=\frac{\text{d}}{\text{d}x}{\color{red}\sec{(x)}}+\frac{\text{d}}{\text{d}x}{\color{red}C}={\color{green}\sec{(x)}\tan{(x)}}+0=f(x)$

Since $\frac{\text{d}}{\text{d}x}F\left(x\right)=f\left(x\right)$$\displaystyle \frac{\text{d}}{\text{d}x}{\color{red}F(x)}=\color{green}f(x)$, then the antiderivative, $F\left(x\right)$$F(x)$, is correct.

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

Warnings:

1. 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.

2. The Mathematica syntax for $\mathrm{tan}\left(x\right)$$\tan{(x)}$ is $\text{Tan[x]}$$\text{Tan[x]}$, ${\mathrm{sec}}^{2}\left(x\right)$$\sec^2{(x)}$ is $\text{Sec[x]}\phantom{\rule{-0.167em}{0ex}}\wedge \phantom{\rule{-0.167em}{0ex}}2$$\text{Sec[x]}\!\wedge\!2$, ${e}^{x}$$e^x$ is $\text{E}\phantom{\rule{-0.167em}{0ex}}\wedge \phantom{\rule{-0.167em}{0ex}}2$$\text{E}\!\wedge\!2$, etc.

3. Integrate[ ] has at least two arguments:

1. The integrand function;

2. The independent variable.

4. For help on using the Integrate[ ] function:

1. In Mathematica, execute the code: $\text{?Integrate}$$\text{?Integrate}$

2. Click on $\vee$$\or$ near the bottom-left of output

3. Click on local

4. Read how to use the Integrate[ ] function – you will be able to copy-paste code.

5. Remember to use parens $\left($$\left(\right.$ and $\right)$$\left.\right)$ to group multiple terms in the numerator or denominator of a rational expression.

6. Remember, correct Mathematica code will be all black except for variables.

7. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

8. Mathematica does not show $+C$$+C$ for antiderivatives. You will need to remember to include $+C$$+C$ in your answers.

9. Mathematica may return a result in a different, but equivalent, form than your answer. In these cases, use another method to check your work.

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:

1. 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.

2. The Python syntax for ${\mathrm{sec}}^{2}\left(x\right)$$\sec^2{(x)}$ is $\text{sec(x)}\phantom{\rule{-0.167em}{0ex}}\ast \phantom{\rule{-0.167em}{0ex}}\ast 2$$\text{sec(x)}\!\ast\!\ast2$, ${e}^{x}$$e^x$ is $\text{exp(x)}$$\text{exp(x)}$, ${x}^{4}$$x^4$ is $\text{x**4}$$\text{x**4}$, etc.

3. Python requires $\ast$$*$ for explicit multiplication, e.g., $\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)$$\sec{(x)}\tan{(x)}$ is $\text{sec(x)*tan(x)}$$\text{sec(x)*tan(x)}$.

4. Remember to use parens $\left($$\left(\right.$ and $\right)$$\left.\right)$​ to group multiple terms in the numerator or denominator of a rational expression.

5. Python does not show $+C$$+C$ for antiderivatives. You will need to remember to include $+C$$+C$ in your answers.

6. Python may return a result in a different, but equivalent, form than your answer. In these cases, use another to check your work.

7. To execute code, press the “View the result” button:

Method 04: Graph in DESMOS

Warnings:

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

2. In Box 2, type in the antiderivative, $F\left(x\right)$$F(x)$. Make sure that you:

• use function notation;

• use an uppercase $F$$F$;

• include $+C$$+C$ [this will create the slider in Box 3];

• click on the color circle in Box 3 to hide the graph of $F\left(x\right)$$F(x)$.

3. In Box 4, type ${\text{F}}^{\phantom{\rule{0.167em}{0ex}}\prime }\text{(x)}$$\text{F}^{\,\prime}\text{(x)}$.

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