Integration by Parts

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

• Section 7.1 – Integration by Parts

Expected Educational Results

• Objective 03–01: I can use integration by parts to evaluate an indefinite integral using FTC-I.

• Objective 03–02: I can use integration by parts to evaluate a definite integral using FTC-II.

• Objective 03–03: I can use substitution and integration by parts to evaluate an indefinite integral using FTC-I.

• Objective 03–04: I can use substitution and integration by parts to evaluate a definite integral using FTC-II.

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.

Integration by Parts

Practice

Practice 14

Evaluate the following integrals:

1. ${\displaystyle \int \frac{\cos (\ln (x))}{x}dx}$$\displaystyle \int{\frac{\cos{\left(\ln{(x)}\right)}}{x}\,dx}$

2. ${\displaystyle \int \cos (\ln (x))dx}$$\displaystyle \int{\cos{\left(\ln{(x)}\right)}\,dx}$

3. ${\displaystyle \int \cos \left(\sqrt{x}\right)dx}$$\displaystyle \int{\cos{\left(\sqrt{x}\right)}\,dx}$

4. ${\displaystyle \int \cos \left(e^x\right)dx}$$\displaystyle \int{\cos{\left(e^x\right)}\,dx}$

5. ${\displaystyle \int \frac{\sin (\ln (x))\cos (\ln (x))}{x}dx}$$\displaystyle \int{\frac{\sin{\left(\ln{(x)}\right)}\cos{\left(\ln{(x)}\right)}}{x}\,dx}$

6. ${\displaystyle \int x\sin \left(x^2\right)dx}$$\displaystyle \int{x\sin{\left(x^2\right)}\,dx}$

7. ${\displaystyle \int x^2\sin \left(3x\right)dx}$$\displaystyle \int{x^2\sin{\left(3x\right)}\,dx}$

8. ${\displaystyle \int x^3\sin \left(x^2\right)dx}$$\displaystyle \int{x^3\sin{\left(x^2\right)}\,dx}$

9. ${\displaystyle \int x^2\ln (x)dx}$$\displaystyle \int{x^2\ln{(x)}\,dx}$

10. ${\displaystyle \int \frac{\ln (x)}{x^3}dx}$$\displaystyle \int{\frac{\ln{(x)}}{x^3}\,dx}$

11. ${\displaystyle \int {\cos }^{-1}\left(x\right)dx}$$\displaystyle \int{\cos^{-1}{\left(x\right)}\,dx}$

12. ${\displaystyle \int x^2{\sin }^{-1}\left(x\right)dx}$$\displaystyle \int{x^2\sin^{-1}{\left(x\right)}\,dx}$

13. ${\displaystyle {\int }_0^1{e}^{2x}\cos \left(e^x\right)dx}$$\displaystyle \int_0^1{e^{2x}\cos{\left(e^x\right)}\,dx}$

14. ${\displaystyle {\int }_0^1{x}^2{e}^{3x}dx}$$\displaystyle \int_0^1{x^2e^{3x}\,dx}$

15. ${\displaystyle {\int }_1^e\ln (x)dx}$$\displaystyle \int_1^e{\ln{(x)}\,dx}$

Use Technology to Verify Antiderivatives

Mathematica

NOTE: Mathematica is used here to verify antiderivatives and definite integrals. You must show that you understand the Calculus concept of the integration.

​x
1
(* Indefinite integral example from Practice 15-1 *)
2
Integrate[Cos[Log[x]]/x, x]
3

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(* Definite integral example from Practice 15-13 *)
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Integrate[E^(2x) Cos[E^x], {x,0,1}]

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 ${\sin }^{-1}(x)$$\sin^{-1}{(x)}$ is $\text{ArcSin[x]}$$\text{ArcSin[x]}$, $e^x$$e^x$ is $\text{E}\wedge x$$\text{E}\!\wedge\!x$, ${\sec }^2(x)$$\sec^2{(x)}$ is $\text{Sec[x]}\wedge 2$$\text{Sec[x]}\!\wedge\!2$​​, etc.

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

4. 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.

5. 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

      1. $x$$x$​ is the variable used in the integrand;

      2. $a$$a$ is the lower limit of the definite integral;

      3. $b$$b$ is the upper limit of the definite integral.

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 FTC-I to check your work.

DESMOS

NOTE: DESMOS is used here to verify definite integrals. On all Assessments, you must show that you understand the Calculus concept of definite integrals.

NOTE: On all Assessments, exact values are expected (unless otherwise directed). DESMOS only returns decimal approximations of definite integrals.

Warnings:

1. At this time, DESMOS cannot evaluate indefinite integrals.

2. The integral symbol can be typeset either by:

   1. typing: int

   2. located under: Show Keypad > functions > Calculus

3. dx is required to return the value of the definite integral.

4. Any variable may be used as long as there is only one variable (including the differential).

Python

NOTE: Python is used here to verify definite integrals. On all Assessments, you must show that you understand the Calculus 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:

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Created: Saturday, 29 January 2022 - 14:07 (EST)