Substitution Rule

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

• Section 5.5 – The Substitution Rule

Prerequisite Knowledge

Algebra

Properties of Exponents

• ${\displaystyle \frac{x^m}{x^n}=x^{m-n}}$$\displaystyle \frac{x^m}{x^n}=x^{m-n}$

• ${\displaystyle \frac{1}{x^n}=x^{-n}}$$\displaystyle \frac{1}{x^n}=x^{-n}$

Rational Exponents

${\displaystyle \sqrt[n]{x^m}=x^{\frac{m}{n}}}$$\displaystyle \sqrt[n]{x^m}=x^{\frac{m}{n}}$

Check Your Work

Use Technology to Express Radicals

Mathematica

​x
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(* Example: Represent cubed root of x^2 *)
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Surd[x^2,3]
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(* The following is NOT equivalent to the above *)
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x^(2/3)

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.

Distributive Property

${\displaystyle x^d(a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n)=a_0{x}^d+a_1{x}^{1+d}+a_2{x}^{2+d}+\cdots +a_n{x}^{n+d}}$$\displaystyle x^d\left(a_0+a_1 x+a_2x^2+\cdots +a_nx^n\right)=a_0x^d+a_1 x^{1+d}+a_2x^{2+d}+\cdots +a_nx^{n+d}$

and

${\displaystyle \frac{a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n}{x^c}=a_0{x}^{-c}+a_1{x}^{1-c}+a_2{x}^{2-c}+\cdots +a_n{x}^{n-c}}$$\displaystyle \frac{a_0+a_1 x+a_2x^2+\cdots +a_nx^n}{x^c}=a_0x^{-c}+a_1 x^{1-c}+a_2x^{2-c}+\cdots +a_nx^{n-c}$

Distributive Property – Table Form

${\displaystyle (a_0+a_1+\cdots +a_n)(b_0+b_1+\cdots +b_n)=b_0{a}_0+b_0{a}_1+\cdots }$$\displaystyle (a_0+a_1+\cdots +a_n)(b_0+b_1+\cdots +b_n)=b_0 a_0 + b_0 a_1 + \cdots$

Distributive Property – Method of Detached Coefficients

Example 01: ${\displaystyle (x^3-2x^2-1)(2x^2+3x+5)}$$\displaystyle (x^3-2x^2-1)(2x^2+3x+5)$

which corresponds to the polynomial $2x^5-1x^4-1x^3-12x^2-3x-5$$2x^5-1x^4-1x^3-12x^2-3x-5$.

Check Your Work

Use Technology to Rewrite Using Algebra

Mathematica

NOTE: On all Assessments, you may use Mathematica to rewrite any expression without showing any work.

x
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(* Example 01: Expand (x^3-2x^2-1)(2x^2+3x+5) *)
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Expand[(x^3-2x^2-1)(2x^2+3x+5)]
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(* Example: Expand (x^2+3x-2)/x *)
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Expand[(x^2+3x-2)/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. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

3. Mathematica may return a result in a different, but equivalent, form than your answer.

Matlab

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

x
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8
11
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% Example 01: Expand (x^3-2x^2-1)(2x^2+3x+5)
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% Declare the independent variable
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syms x
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% Integrate the function
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expand((x^3-2*x^2-1)*(2*x^2+3*x+5))
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% Example 02: Expand (x^2+3x-2)/x
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expand((x^2+3x-2)/x)

Link to Mathematica Notebook file containing the above code: http://calc.jjw3.com/math2212/nb/02d_Indefinite_Integrals.nb.

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. expand( ) for has at least one argument:

   1. The expression to be simplified.

3. The Matlab syntax for $e^x$$e^x$​ is $\text{exp}(x)$$\text{exp}(x)$​; $\cos (x)$$\cos{(x)}$​ is $\text{cos(x)}$$\text{cos(x)}$​, $\ln (x)$$\ln{(x)}$​ is $\text{log(x)}$$\text{log(x)}$​, etc.

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

5. Matlab requires $\ast$$*$ for explicit multiplication.

Python

NOTE: On all Assessments, you may use Python to rewrite any expression without showing any work.

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. Python requires $\ast$$*$ for explicit multiplication.

3. Python requires $\ast \ast$$**$ for exponentiation, e.g., $x^3$$x^3$ is $\text{x**3}$$\text{x**3}$.

4. Remember to use parens $($$\left(\right.$ and $)$$\left.\right)$​ to group factors consisting of multiple terms.

5. You can use pprint( ) for pretty print.

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

Properties of Logarithmic Functions

The only property of logarithmic functions needed for $u$$u$-Substitution:

• ${\displaystyle {\log }_b\left(x^n\right)=n{\log }_b\left(x\right)}$$\displaystyle \log_b{\left(x^n\right)}=n\log_b{\left(x\right)}$, where $x>0$$x>0$

Check Your Work

Use Technology to Rewrite Using Algebra

Mathematica

NOTE: On all Assessments, you may use Mathematica to rewrite any expression without showing any work.

NOTE: This line of Mathematica code rewrite a log expression using the Power Property of Logarithms.

xxxxxxxxxx
2
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(* Simplify: ln(x^3) *)
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FullSimplify[Log[x^3], x>0]

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 argument $\text{x>0}$$\text{x>0}$ forces Mathematica to use Real-values answers.

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

CC BY-NC-SA 4.0

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: Monday, 18 January 2021 09:05 EDT Last Modified: Monday, 08 May 2023 - 10:59 (EST)