Integration By Parts
Overview: In this lesson, students will be introduced to another method of integration,
integration by parts. While substitution helps us recognize derivatives produced by the
chain rule, integration by parts helps us to recognize derivatives produced by the product
rule.
Grade Level/Subject: The lesson is for 12th graders in AP Calculus.
Time: 1-50 minute class period
Purpose: This lesson will give students yet another method of solving complicated
integrals. It will also give them a small introduction into differential equations.
Prerequisite Knowledge:
Student should:
- Understand Definite Integrals
- Understand and be able to perform Integration by Substitution
- Know the product rule for derivatives
Objectives:
1. Students will learn to do integration by parts and be able to use the method
comfortably.
2. Students will learn how do decide when to use integration by substitution and
when to use integration by parts.
Standards:
1. Problem-Solving: Students will use their problem-solving techniques to determine
which method to use for particular integrals.
2. Communications: Students will have to communicate clearly what method is
being used and all the steps taken to achieve their answer.
3. Connections: Students will need to recall the product rule from differential
calculus to recognize integrals that can be solved through integration by parts.
Resources/Materials Needed:
1. Calculus Book
2. Dry Erase Board/Dry Erase Markers
3. Attached Handouts and Worksheets
Activities and Procedures:
1. Begin by having the students recall the product rule from differential calculus.
d
( f ( x) g ( x)  f ' ( x) g ( x)  f ( x) g ' ( x)
dx
If we write u for f(x) and v for g(x), we have
d
du
dv
(uv) 
vu
dx
dx
dx
From here, we can reverse the product rule:
uv   vdu   udv
When we subtract  vdu from both sides, we get the integration by parts formula.
 udv  uv   vdu
2. Give the steps for using Integration by Parts.
Step 1: Let u = f(x) and dv = g(x)dx, where f(x)g(x)dx is the original integrand.
Good choices to make are integrals  dv   g ( x)dx , which are easy to integrate.
Step 2: Compute du = f’(x)dx and v =
 g ( x)dx .
Step 3: Substitute u, v, du, and dv into the formula,  udv  uv   vdu
Step 4: Calculate uv -  vdu
If  vdu is difficult or impossible to integrate, go back to step 1 and consider other
choices for u and dv.
Step 5: Check your solution by differentiating and comparing to the original
integrand.
3. Do several examples to demonstrate the method. Have the students do one within
their groups and one on their own. The best way to learn this method is to
practice, practice, practice!
a.  xe x dx
b.
c.
d.
 x e dx
 ln( x)dx
 arctan( x)dx
2
x
4. Give the two attached Handouts to the students to help them.
Homework: Read Section 6.9 about improper integrals and take notes. Complete the
attached Worksheet.
1. Let u = f(x) and dv = g(x)dx, where f (x)g(x)dx
is the original integrand.
 Try making dv be the most complicated
portion of the integrand that fits a basic
integration formula. Then u will be the
remaining factor(s) of the integrand.
 Try letting u be the portion of the integrand
whose derivative is a simpler function and
then dv will be the remaining factor(s) of
the integrand.
2. Compute du = f ‘ (x) dx and v   g (x )dx .
3. Substitute u, v, du, and dv into the formula
 udv  uv  vdu
4. Evaluate uv   vdu . (If the integral is difficult
or impossible to integrate, go back to Step 1 and
consider other choices for u and dv.
5. Check your solution by differentiating and
comparing it to the original integrand.
Summary of Common Integrals
using Integration by Parts
1. For integrals of the form
n ax
x
 e dx ,
n
n
x
sin(
ax
)
dx
,
or
x

 cos(ax )dx
let u = xn
let dv = eax or sin(ax)dx, or cos(ax)dx
2. For integrals of the form
n
x
 ln(x )dx ,
n
x
 arcsin(ax )dx , or
n
x
 arctan(ax )dx
let u = ln(x), arcsin(ax), or arctan(x)
let dv = xndx
3. For integrals of the form
ax
e
 sin(bx )dx
or
ax
e
 cos(bx )dx
let u = sin(bx) or cos(bx)
let dv = eaxdx
Name: __________________
1.
2.
3.
4.
 cos  2 x  e
 xe
x
 5ne
3x
dx
x
10.
x
11.
 arccos x
dx
3n
dn
 5x sin 3x  dx
5.
x
 3x e dx
6.
ln  x dx
9.
2
2
cos x dx
12.  cos
4

3
1  x 2 dx

2x  3 dx
13.
 7  3x  e
 arcsin(3x )dx
14.
 x
7.
 x arctan  x dx
15.
 3  5x  cos  4x  dx
8.
x
16.
 x ln  x  dx
2 x
e dx
3
6x
dx

 5 ln  x dx
1