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barrels
200
Problem of the Day
100
6
12
18
24
hours
The flow of oil, in barrels per hour, through a pipeline on
July 9 is given by the graph above. Of the following, which
best approximates the total number of barrels of oil that
passed through the pipeline that day?
A) 500 B) 600 C) 2,400
D) 3,000
E) 4,800
barrels
200
Problem of the Day
100
6
12
18
24
hours
The flow of oil, in barels per hour, through a pipeline on
July 9 is given by the graph above. Of the following, which
best approximates the total number of barrels of oil that
passed through the pipeline that day?
A) 500
4,800
B) 600
C) 2,400
D) 3,000
E)
Homework Questions?
4-5: Integration By
Substitution
Objectives:
•Learn and practice the
substitution technique for
integration
•Integrate even and odd
functions
©2003 Roy L. Gover (www.mrgover.com)
Review
Find the derivative:
1. f ( x)  2 x
2
2. g ( x)  5sin 5 x
3. y  (2 x  7)
3
Important Idea
•The role of substitution in
integration is comparable to the
chain rule in differentiation.
•Substitution, sometimes called
u-substitution (u-sub), allows
integration that otherwise could
not be done.
Example
Find the
antiderivative:
u
du
2x
(
x

1)
(2
x
)
dx
 du
du
Then:
Let:

2
x
2
And: du
u 
x

1
2
xdx

dx

dx
2x
2
2
Procedure
•Let u=the inside function.
•Find du in terms of dx.
•Substitute and find the
antiderivative in terms of u.
•Substitute the expression
for x in place of u.
Example
Find the antiderivative:
u
5cos
5xdx

Try This
Find the antiderivative using
a u-substitution then
differentiate your result:
sin
2xdx

1
 cos(2 x)  c
2
Warm-Up
Solve the differential
equation with the initial
condition of y(0)=2:
y (t )   [t sin(t )]dt
2
1
5
2
y (t )   cos(t ) 
2
2
Example
Find the antiderivative
x
(
x

1)
dx

2
2
Important Idea
The u-substitution must
eliminate any product of
variables.

du 2
xu
x( x 21)
dx
x
2
2
Example
Find the antiderivative:

2 x  1dx

2 x  1dx
Example
Find the antiderivative:
x
2
x

1
dx

This example requires a
slightly different technique:
x
2
x

1
dx

Try This
Find the antiderivative:
(
x

1)
2

xdx

3
2
5
2
2
2(2  x)  (2  x)  c
5
(
x

1)
2

xdx

Example
Find the antiderivative:
sin
3
x
cos
3
xdx

2
Try This
Find the antiderivative:
tan
x
sec
xdx

2
2
1
3
tan x  c
3
Try This
Find the antiderivative:
3(3
x

1)
dx

4
1
5
(3 x  1)  c
5
Example
Find the antiderivative:
Let u=?
(2
x

1)(
x

x
)
dx

2
Example
Find the antiderivative:
(3
x

2
x  2)dx
3
Try This
Find the antiderivative:
cos
x
sin
xdx

2
3
cos x

c
3
Assignment
1. 304/7-25 odd & 3137odd(find the general
solution), 43-49 odd
Slides 1-21
2. 305/51-63 odd ,71,73
Slides 22-38
Warm-Up
Find the antiderivative:
4 x
dx
 (1  2 x 2 )2
1


c
2
1 2x
Important Idea
Substitution methods are
also used with definite
integrals. There are two
methods:
•Reverse the substitution
and use original limits
•Change the variable and
use new limits
Evaluate:
Example
1
x
(
x

1)
dx

2
0
3
Try This
2
(
x

1)
dx

7
0
820
Example
Evaluate:
5

1
x
dx
2x 1
Definition
Odd functions are
functions symmetric with
the origin. They have
exponents that are all
odd.
Example
f ( x) 
sin x cos x  sin x cos x
3
Odd
Function
Definition
Even Functions are
functions symmetric with
the y axis. They have
exponents that are all
even. Constants are
considered to have even
exponents.
Example
f ( x)  x  2 x  1
4
Even
Function
2
Try This
Even, odd or neither?
1. f ( x)  3x  2
4
2. g ( x)  x  x
3
3. p( x)  sin x  cos x
2
Important Idea
If f is an even function:

a
a
a
f ( x)dx  2 f ( x)dx
0
Important Idea
If f is an odd function:

a
a
f ( x)dx  0
Example
Evaluate:
 2


(sin x cos x  sin x cos x)dx
3
2
Example
Evaluate:
1
(
x

2
x

2)
dx

4
1
2
Try This
Evaluate:

2
2
( x  x)dx
3
0
Lesson Close
Describe generally how
you integrate a function
by substitution.
Assignment
1. 304/7-25 odd & 3137odd(find the general
solution), 43-49 odd
Slides 1-21
2. 305/51-63 odd ,71,73
Slides 22-38