# 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.

## Indefinite Integrals

### Definition: Indefinite Integral

An Indefinite Integral has the form: $\int f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\int{f(x)\,dx}$

The result when evaluating an Indefinite Integral is $\int f\left(x\right)\phantom{\rule{0.167em}{0ex}}\text{d}x=F\left(x\right)+C$$\int{f(x)\,\text{d}x}=F(x)+C$ if and only if ${F}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)=f\left(x\right)$$F^{\,\prime}(x)=f(x)$.

Details of the Definition of Indefinite Integrals:

• ${F}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)=f\left(x\right)$$F^{\,\prime}(x)=f(x)$ means that $f\left(x\right)$$f(x)$ is a derivative function.

• $\int f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\int{f(x)\,dx}$ means adding values of the derivative function $f\left(x\right)$$f(x)$.

• We will describe this in more detail soon.

• The sum of $f\left(x\right)\phantom{\rule{0.167em}{0ex}}\text{d}x$$f(x)\,\text{d}x$ values results in the $F\left(x\right)$$F(x)$ function.

• $F\left(x\right)+C$$F(x)+C$ is called the antiderivative of $f\left(x\right)$$f(x)$.

NOTE:

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

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

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

The phrase if and only if is sometimes abbreviated iff.

The mathematical representation of the phrase if and only if is the symbol $⟺$$\Longleftrightarrow$.

#### Notation for Indefinite Integrals

$\int$$\int$ will be used for sums, i.e., indefinite integrals, using continuous functions whereas $\sum$$\sum$ is used for sums using discrete functions (we will see this later in the course).

### Evaluating Indefinite Integrals

#### Rewrite the Indefinite Integral

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.

#### Investigation 03

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).

1. $\int \left({x}^{2}+\sqrt[3]{x}\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\left(x^2+\sqrt[3]{x}\right)\,dx}$

2. $\int \left({x}^{2}-2x+1\right)\left(1-3x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\left(x^2-2x+1\right)\left(1-3x\right)\,dx}$

3. $\int \sqrt[3]{x}\left(1+x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\sqrt[3]{x}\left(1+x\right)\,dx}$

4. $\int \frac{3{x}^{2}+5x-2}{x}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{3x^2+5x-2}{x}\,dx}$

5. $\int \frac{\left(x+1{\right)}^{3}}{x}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{(x+1)^3}{x}\,dx}$

6. $\int \frac{x-3\sqrt{x}+1}{\sqrt{x}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x-3\sqrt{x}+1}{\sqrt{x}}\,dx}$

7. $\int \frac{{x}^{2}-x+1}{{x}^{2}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x^2-x+1}{x^2}\,dx}$

8. $\int \frac{1}{{x}^{2}+1}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{1}{x^2+1}\,dx}$

9. $\int \left({x}^{-3/4}+{x}^{1/4}-{e}^{x}\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\left(x^{-3/4}+x^{1/4}-e^x\right)\,dx}$

10. $\int \frac{-1}{\sqrt{1-{x}^{2}}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{-1}{\sqrt{1-x^2}}\,dx}$

11. $\int \frac{\mathrm{sin}\left(x\right)}{{\mathrm{cos}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{\sin{(x)}}{\cos^2{\!(x)}}\,dx}$

12. $\int \frac{\mathrm{sin}\left(2x\right)}{\mathrm{cos}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{\sin{(2x)}}{\cos{(x)}}\,dx}$

13. $\int \frac{\mathrm{sin}\left(2x\right)}{\mathrm{sin}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{\sin{(2x)}}{\sin{(x)}}\,dx}$

14. $\int \frac{\mathrm{sec}\left(x\right)}{\mathrm{cos}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{\sec{(x)}}{\cos{(x)}}\,dx}$

15. $\int \left[{\mathrm{cos}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(\frac{x}{2}\right)-{\mathrm{sin}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(\frac{x}{2}\right)\right]\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\left[\cos^2{\!\left(\frac{x}{2}\right)}-\sin^2{\!\left(\frac{x}{2}\right)}\right]\,dx}$

16. $\int {\left[{\mathrm{sin}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(x\right)+{\mathrm{cos}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(x\right)\right]}^{3}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\left[\sin^2{\!(x)}+\cos^2{\!(x)}\right]^3\,dx}$

#### Investigation 04

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

#### Use Technology to Rewrite Using Algebra and Trigonometry

Mathematica

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

​x1(* Simplify: (x^2-x+2)/x *)2Simplify[(x^2-x+2)/x]3
4(* Simplify: (x^2-x+2)/x *)5FullSimplify[(x^2-x+2)/x]6
7(* Expand: (x^2-x+2)(x+1) *)8Expand[(x^2-x+2)(x+1)]9
10(* Rewrite: sin(x)/(cos(x))^2 *)11TrigExpand[Sin[x]/Cos[x]^2]12
13(* Rewrite: (cos(x))^2 *)14TrigReduce[Cos[x]^2]

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. Do NOT memorize the specifics of each of the above commands; know the commands and try them until the result is what you need.

4. All of the commands shown above have only one (1) argument:

1. The expression to be simplified, if possible.

5. For help on using the above commands, e.g., FullSimplify[ ] function:

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

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

3. Click on local

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

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

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

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

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

#### 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.

xxxxxxxxxx41(* 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.

### Homework

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.