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.

Practice

Practice 23

Show ALL work. Evaluate the following definite integrals:

1. ${\displaystyle {\int }_{-1}^{2}x(x+5{)}^7\text{d}x}$$\displaystyle \int_{-1}^2{x(x+5)^7\,\text{d}x}$

2. ${\displaystyle {\int }_2^{4}x(x^2-1{)}^8\text{d}x}$$\displaystyle \int_2^4{x(x^2-1)^8\,\text{d}x}$

3. ${\displaystyle {\int }_1^2(x+3)(x-1{)}^5\text{d}x}$$\displaystyle \int_1^2{(x+3)(x-1)^5\,\text{d}x}$

4. ${\displaystyle {\int }_0^{2}x{e}^{x^2}\text{d}x}$$\displaystyle \int_0^2{x e^{x^2}\,\text{d}x}$

5. ${\displaystyle {\int }_0^{\pi /4}\sin (5x)\text{d}x}$$\displaystyle \int_0^{\pi/4}{\sin{(5x)}\,\text{d}x}$

6. ${\displaystyle {\int }_1^3\frac{6x}{1+3x^2}\text{d}x}$$\displaystyle \int_1^3{\frac{6x}{1+3x^2}\,\text{d}x}$

7. ${\displaystyle {\int }_0^3{e}^{-x}\text{d}x}$$\displaystyle \int_0^3{e^{-x}\,\text{d}x}$

8. ${\displaystyle {\int }_1^4\frac{\cos \left(\sqrt{x}\right)}{\sqrt{x}}\text{d}x}$$\displaystyle \int_1^4{\frac{\cos{\left(\sqrt{x}\right)}}{\sqrt{x}}\,\text{d}x}$

9. ${\displaystyle {\int }_1^{4}x\cos \left(x^2\right)\text{d}x}$$\displaystyle \int_1^4{x\cos{\left(x^2\right)}\,\text{d}x}$

10. ${\displaystyle {\int }_1^3\frac{\ln \left(x\right)}{x}\text{d}x}$$\displaystyle \int_1^3{\frac{\ln{\left(x\right)}}{x}\,\text{d}x}$

11. ${\displaystyle {\int }_1^3\frac{\ln \left(x^2\right)}{x}\text{d}x}$$\displaystyle \int_1^3{\frac{\ln{\left(x^2\right)}}{x}\,\text{d}x}$

12. ${\displaystyle {\int }_{-1}^{2}x\sqrt{x^2+2}\text{d}x}$$\displaystyle \int_{-1}^2{x\,\sqrt{x^2+2}\,\text{d}x}$

13. ${\displaystyle {\int }_{-1}^2\frac{x}{\sqrt{x^2+2}}\text{d}x}$$\displaystyle \int_{-1}^2{\frac{x}{\sqrt{x^2+2}}\,\text{d}x}$

14. ${\displaystyle {\int }_{-1}^2{x}^3\sqrt{x^2+2}\text{d}x}$$\displaystyle \int_{-1}^2{x^3\,\sqrt{x^2+2}\,\text{d}x}$

15. ${\displaystyle {\int }_0^2{e}^x\ln (e^x+2)\text{d}x}$$\displaystyle \int_0^2{e^x\,\ln{\left(e^x+2\right)}\,\text{d}x}$

Use Technology to Verify Definite Integrals

Mathematica

NOTE: Mathematica is used here to verify definite integrals. On all Assessments, you must show that you understand the Calculus II concept of using $u$$u$​-Substitution to evaluate a definite integral. On all Assessments, answers are expected to be in exact values (not decimal approximations), except when directed otherwise.

​x
1
(* Investigation 14-6 *)
2
(* Evaluate indefinite integral cos(sqrt(x^2))/sqrt(x) from x=1 to x=4 *)
3

4
Integrate[Cos[Sqrt[x^2]]/Sqrt[x], {x,1,4}]

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 definite integrals has two arguments:

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

   2. $\{x,a,b\}$$\{x,a,b\}$, where $x$$x$​ is the variable used in the integrand, $a$$a$ is the lower limit of integration; $b$$b$ is the upper limit of integration.

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 definite integrals. On all Assessments, you must show that you understand the Calculus II concept of using $u$$u$-Substitution to evaluate a definite integral. On all Assessments, answers are expected to be in exact values (not decimal approximations), except when directed otherwise.

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 definite integrals has two arguments:

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

   2. $(x,a,b)$$(x,a,b)$, where $x$$x$ is the variable used in the integrand, $a$$a$ is the lower limit of integration; $b$$b$ is the upper limit of integration.

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 be able to complete the following assignments:

• Section 5.5: # 1--27 (odd), 29--47(odd), 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73.

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: Tuesday, 25 August 2020 05:33 EDT Last Modified: Monday, 23 August 2021 - 06:20 (EDT)