Substitution Rule

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

• Section 5.5 – The Substitution Rule

Expected Educational Results

• Objective 02–01: I can use a substitution to evaluate an indefinite integral using FTC-I.

• Objective 02–02: I can use a substitution to evaluate a definite integral.

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.

Special Case for Substitution Methods

Linear “Inside” Functions

Example 07: Consider the integral ${\displaystyle \int e^{3x}\text{d}x}$$\displaystyle \int{e^{3x}\,\text{d}x}$.

The “inside” function is the linear function, $3x$$\color{red}3x$.

NOTE: Suppose we wanted to differentiate ${\displaystyle \frac{d}{dx}{e}^{3x}}$$\displaystyle \frac{d}{dx}\,e^{3x}$

We would use the Chain Rule for Derivatives:

${\displaystyle \frac{d}{dx}{e}^{3x}=e^{3x}\frac{d}{dx}3x=3e^{3x}}$$\displaystyle \frac{d}{dx}\,e^{3x}=e^{3x}\frac{d}{dx}{\color{red}3x}={\color{red}3}\,e^{3x}$

where we multiplied by the derivative of the “inside” function.

Answer 07:

Recall, FTC-I states that the process of integrating and the process of differentiating are inverse operations.

Thus, if we multiplied by the derivative of the ‘inside’ function when differentiating, then we divide by the derivative of the “inside” function when integrating!

So, ${\displaystyle \int e^{3x}\text{d}x=\frac{e^{3x}}{3}+C}$$\displaystyle \int{e^{3x}\,\text{d}x}=\frac{e^{3x}}{\color{red}3}+C$

Check:

${\displaystyle \frac{d}{dx}(\frac{e^{3x}}{3}+C)=\frac{\cancel{3}{e}^{3x}}{\cancel{3}}+0=e^{3x}}$$\displaystyle \frac{d}{dx}\left(\frac{e^{3x}}{\color{red}3}+C\right)=\frac{\cancel{\color{red}3}e^{3x}}{\cancel{\color{red}3}}+0=e^{3x}$

Example 08: Evaluate: ${\displaystyle \int \sin \left(\frac{x}{2}\right)\text{d}x}$$\displaystyle \int{\sin{\left(\frac{x}{2}\right)}\,\text{d}x}$.

Answer 08:

The “inside” function is ${\displaystyle \frac{x}{2}}$$\displaystyle\frac{x}{2}$ which is linear and the derivative is ${\displaystyle \frac{d}{dx}\left(\frac{x}{2}\right)=\frac{1}{2}}$$\displaystyle\frac{d}{dx}\left(\frac{x}{2}\right)=\frac{1}{2}$.

So, ${\displaystyle \int \sin \left(\frac{x}{2}\right)\text{d}x=-\frac{\cos \left(\frac{x}{2}\right)}{\frac{1}{2}}+C}$$\displaystyle \int{\sin{\left(\frac{x}{2}\right)}\,\text{d}x}=-\frac{\cos{\left(\frac{x}{2}\right)}}{\frac{1}{2}}+C$.

Check:

${\displaystyle \frac{d}{dx}(-\frac{\cos \left(\frac{x}{2}\right)}{\frac{1}{2}}+C)=\bcancel{-}\frac{\bcancel{-}\cancel{\frac{1}{2}}\sin \left(\frac{x}{2}\right)}{\cancel{\frac{1}{2}}}+0=\sin \left(\frac{x}{2}\right)}$$\displaystyle \frac{d}{dx}\left(-\frac{\cos{\left(\frac{x}{2}\right)}}{\frac{1}{2}}+C\right)=\bcancel{-}\frac{\bcancel{-}\cancel{\frac{1}{2}}\,\sin{\left(\frac{x}{2}\right)}}{\cancel{\frac{1}{2}}}+0=\sin{\left(\frac{x}{2}\right)}$

Investigation 17

Evaluate the following:

1. ${\displaystyle \int e^{-x}\text{d}x}$$\displaystyle \int{e^{-x}\,\text{d}x}$

2. ${\displaystyle \int {\sec }^2(5x)\text{d}x}$$\displaystyle \int{\sec^{2}{\!(5x)}\,\text{d}x}$

3. ${\displaystyle \int \frac{1}{3x+7}\text{d}x}$$\displaystyle \int{\frac{1}{3x+7}\,\text{d}x}$

4. ${\displaystyle \int \cos (2-3x)\text{d}x}$$\displaystyle \int{\cos{\!(2-3x)}\,\text{d}x}$

5. ${\displaystyle \int (4x-9{)}^8\text{d}x}$$\displaystyle \int{(4x-9)^8\,\text{d}x}$

Use Technology to Verify Indefinite Integrals

Mathematica

NOTE: Mathematica is used here to verify indefinite integrals. On all Assessments, you must show that you understand the Calculus concept of using $u$$u$​-Substitution to evaluate an indefinite integral.

​x
1
(* Example 20-4 *)
2

3
Integrate[Cos[2-3x], 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. Integrate[ ] for indefinite integrals has two arguments:

   1. The integrand, i.e., function of the definite integral;

   2. $x$$x$​, where $x$$x$​ is the variable used in the integrand.

3. The Mathematica syntax for $e^x$$e^x$​​ is $\text{E}\wedge x$$\text{E}\!\wedge\!x$​​; $\cos (x)$$\cos{(x)}$​​ is $\text{Cos[x]}$$\text{Cos[x]}$​​, $\ln (x)$$\ln{(x)}$​​ is $\text{Log[x]}$$\text{Log[x]}$​​, ${\tan }^{-1}(x)$$\tan^{-1}{(x)}$​​ is $\text{ArcTan[x]}$$\text{ArcTan[x]}$​​, etc.

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

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

6. Mathematica does not return the antiderivative with $+C$$+C$​. You need to use $+C$$+C$ for all steps that are antiderivatives.

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

Python

NOTE: Python is used here to verify indefinite integrals. On all Assessments, you must show that you understand the Calculus I concept of using $u$$u$-Substitution to evaluate an indefinite integral.

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. All code is included above with comment codes that explain the executable code.

3. integrate( ) for indefinite integrals has two arguments:

   1. The integrand, i.e., function of the definite integral;

   2. $x$$x$​, where $x$$x$​​ is the variable used in the integrand.

4. You only need to change the function in the integrate( ) command.

5. The Python syntax for $e^x$$e^x$​​ is $\text{exp(x)}$$\text{exp(x)}$​​; ${\cos }^2(x)$$\cos^2{(x)}$​​ is $\text{cos(x)**2}$$\text{cos(x)**2}$​​; $x^4$$x^4$​​ is $\text{x**4}$$\text{x**4}$​​, $\ln (x)$$\ln{(x)}$​​ is $\text{log(x)}$$\text{log(x)}$​​, ${\tan }^{-1}(x)$$\tan^{-1}{(x)}$​​ is $\text{atan(x)}$$\text{atan(x)}$​​, etc.

6. Python requires $\ast$$*$ for explicit multiplication; $\ast \ast$$**$​ for exponents.

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

8. Python does not return the antiderivative with $+C$$+C$. You need to use $+C$$+C$ for all steps that are antiderivatives.

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

Homework

At this time, you should retry the following assignments using the method discussed on this page:

• Section 5.5: # 1, 13, 15, 39, 59, 61.

CC BY-NC-SA 4.0

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Created: Sunday, 30 January 2022 - 11:41 (EST) Last Modified: Tuesday, 16 August 2022 - 10:42 (EDT)