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 for Definite Integrals

Definition: Integration by Parts

If $U=f(x)$$U=f(x)$ is a differentiable function and $V=g(x)$$V=g(x)$ is a integrable function and continuity conditions on $[a,b]$$[a, b]$ are met, then $dU=f^{\prime }(x)$$dU=f^{\,\prime}(x)$ and $dV=g^{\prime }(x)$$dV=g^{\,\prime}(x)$. Thus,

${\int }_a^{b}UdV={\left(UV\right)|}_a^b-{\int }_a^{b}VdU$$\int_a^b{U\,dV}=\left.\left(UV\right)\right|_a^b-\int_a^b{V\,dU}$

Example 07: Evaluate ${\displaystyle {\int }_0^{2}x{e}^{x}dx}$$\displaystyle \int_0^2{xe^x\,dx}$

Solution 07:

${\displaystyle {\int }_0^{2}x{e}^{x}dx={\left(x{e}^x\right)|}_0^2-{\int }_0^2{e}^{x}dx}$$\displaystyle \int_0^2{xe^x\,dx} = {\color{red}\left.\left(xe^x\right)\right|_0^2}-{\color{blue}\int_0^2{e^x\,dx}}$

${\displaystyle \Rightarrow {\left(x{e}^x\right)|}_0^2-{e^x|}_0^2=(2e^2-0)-(e^2-e^0)=e^2+1}$$\displaystyle \Rightarrow \left.\left(xe^x\right)\right|_0^2-\left.e^x\right|_0^2 = (2e^2 - 0) - (e^2 - e^0) = e^2+1$

NOTE:

• The red term, ${\left(x{e}^x\right)|}_0^2$${\color{red}\left.\left(xe^x\right)\right|_0^2}$, shown in the first line of the solution above is equivalent to a number;

• The blue integral, ${\int }_0^2{e}^{x}dx$${\color{blue}\int_0^2{e^x\,dx}}$, shown in the first line of the solution above is a definite integral and is equivalent to a number;

• Since the red term and the blue integral are both equivalent to numbers, they can be combined by subtraction.

Practice 13

Integrate the following:

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

2. ${\displaystyle {\int }_0^{\pi }x\cos (x)dx}$$\displaystyle \int_0^{\pi}{x\cos{(x)}\,dx}$​​​​

3. ${\displaystyle {\int }_{-1}^1{e}^{6x}\sin (e^{2x})dx}$$\displaystyle \int_{-1}^{1}{e^{6x}\sin{(e^{2x})}\,dx}$​​

4. ${\displaystyle {\int }_1^{e}x{(\ln (x))}^2}$$\displaystyle \int_1^e{x\left(\ln{(x)}\right)^2}$​

5. ${\displaystyle {\int }_0^{1}x{e}^{-x}dx}$$\displaystyle \int_0^1{xe^{-x}\,dx}$

Use Technology to Verify Definite Integrals

Mathematica

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

1
(* Example from Example 04: Integrate x^2e^x *)
2
Integrate[x*E^x, {x, 0, 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 ${\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$​​, $\pi$$\pi$ is $\text{Pi}$$\text{Pi}$, 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,0,2\}$$\{x, 0, 2\}$, where $x$$x$​​ is the variable used in the integrand, $0$$0$ is the lower limit of integration, $2$$2$ is the upper limit of integration.

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

6. 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 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,0,2)$$(x, 0, 2)$​​, where $x$$x$​​​ is the variable used in the integrand, $0$$0$ is the lower limit of integration, $2$$2$ 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)}$​​, $\pi$$\pi$ is $\text{pi}$$\text{pi}$, 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. To execute code, press the “View the result” button: ​​

Homework

At this time, you should be able to complete the following assignments:

• Section 7.1: # 33, 37, 45.

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: Thursday, 12 August 2021 - 15:46 (EDT) Last Modified: Saturday, 04 June 2022 - 19:39 (EDT)